Following up on Edwin Ashworth's very useful wiki answer, I note that Early English Books Online finds five instances of unballanced from the 1640s (the earliest decade of occurrence cited by Etymology Online), but also three from the 1630s. Because the context of the earliest uses is of particular interest in this case, I reproduce all eight instances below. (The earliest EEBO match for unbalanced spelled with just one l is from 1652.)
From Thomas Jackson, The Humiliation of the Sonne of God by His Becomming the Son of Man, by Taking the Forme of a Servant, and by His Sufferings under Pontius Pilat, &c. Or The Eighth Book of Commentaries vpon the Apostles Creed (1635):
But was it possible that either the collapsed Angels, or man by their suggestion, should attempt or desire to bee equall with God, or to bee Gods Almighty? To bee in all points coequall with God, was perhaps more than Lucifer himselfe did desire: yet that even our first Parents desired to bee in some sort or other equall with God, is probable from the Apostles character of the Sonne of God [Hee being, saith hee, in the forme of God, thought it no robbery to bee equall with God.] This to my understanding implies, that the robbery or sacriledge committed by our first Parents for which the Sonne of God did humble and ingage himselfe to make satisfaction, was their proud or haughty attempt to be equal with God, at lest in knowledge of good and evill. And yet, as was said before, the collapsed Angels had doubtlesse sinned more presumptuously, before they tempted our first Parents to the like sinne. Neither man nor Angel could have affected equality in any one attribute with their Creator, much lesse in all or most, so they had made his glory, power, or majesty, the chiefe or principall object of their first contemplations. But how farre the previal sinne of omitting this duty, might let loose their strong and swift imaginations unballanced with experience, or what entrance it might work for that desperate and positive sinne of Ambition, or seeking to bee equall or like to God for power and wisdome; God, and they onely know, if haply they now know, or perfectly remember the maner of their first transgressions. Many things, many learned and wise men doe, and attempt more, through incogitancy, want of consideration (or ad pauca respicientes) which by men of meaner parts would bee suspected for a spice of madnesse, if they had taken them into serious consideration before.
This first instance corroborates Etymology Online's contention that "Earliest use is in reference to the mind, judgment, etc." Not so the second occurrence, however.
From Adam Harsnett, A Cordiall for the Afflicted Touching the Necessitie and Utilitie of Afflictions. Proving unto Us the Happinesse of Those That Thankfully Receive Them: and the Misery of All That Want Them, or Profit Not by Them (1638):
Wee enjoy and command our selves; for impatience puts a man out, and makes to be beside himselfe. By faith wee possesse Christ, by love wee possesse our neighbor, yea our enemie, and by patience wee possesse our selves. He hath but a weak hold of Christ, or of his neighbor, that hath no hold, or command of himselfe. An impatient person is as one out of the way, or as a bone dislocated, and out of joynt. What stabilitie can be, where Patience sits not at the stern to direct and govern? A ship that rides at sea well ballanced, is steddy, and so proves comfortable unto the Passengers that bee abord her: whereas an unballanced vessell reels (like a drunken man) and tumbles too and fro with every little gale, and blast of wind, and so make those weary, if not sick, that be in her.
Here the illustrative comparison to an unstable mind is "an unbalanced vessell"—that is, a ship that has been poorly loaded and is therefore at greater than usual danger of violently pitching or even capsizing in adverse conditions. This would seem to contradict Etymology Online's assertion, "Of material things, it [unbalanced] is recorded from 1732."
From William Sclater, "The Worthy Communicant Rewarded. Laid Forth in a Sermon, on John 6.54. Preached in the Cathedrall of St. Peter in Exeter, on Low-Sunday, Being the 21. of Aprill, Anno 1639" (1639):
And secondly, if they would follow learned Hookers counsell, a worthy instrument in our Church, who wished that men would more give themselves to meditate with silence [what] wee have by the Sacrament, and lesse to dispute of the [manner] (how,) at least considering that successe which Truth hath hitherto had by so bitter conflicts with errour in this point: Thus if we could be perswaded, oh what honey might we sucke as a Samson from his Lyon, from this blessed Sacrament, for our peace and comfort, which now those bitter waters of Meribah, and strife, running downe so violently in a floud, doe, in a sort, wash off from many a seduced and unballanced soule! But woe, and alas! how may our mother the Church, well typed in the Arke of Noah, (she is so tossed on the working billowes of windy, yet boysterous spirits) speake out with Rebekkah, when shee felt her Twinnes to struggle together within her, If it bee so, that I have conceived,Why am I thus? what meanes this strange, and this unnaturall elbowing, and shouldring, and justling together in the same womb betweene Brethren?
This example is interesting because, although Sclater speaks specifically of an "unballanced soule," he describes such souls as being swept away in a flood, and then he turns his attention to Noah's Ark as a symbol for "our mother the Church," which alone preserves souls from destruction.
From Daniel Rogers, Naaman the Syrian His Disease and Cure Discovering Lively to the Reader the Spirituall Leprosie of Sinne and Selfe-Love, Together with the Remedies, viz. Selfe-Deniall and Faith (1642):
Nourish humility and simplicity of spirit (next to faith) above all: Inure thy selfe to deny much for God, that so he may grow dearer to thee, and thou to him: Spend not, nor waste thy zeale needlessely and rashly upon objects of lesser weight, but reserve thy selfe, till a better warrant, and call, a stronger, and more weighty object to pull thee forth: lest thou faile in the hottest of the attempt, as those mostly doe, whose zeale is unballanced. Be not wedded to thy selfe: for the spirit of grace doth not so well befit him, who abounds in his owne sense. Moses was a man who in his own matters was very meek and calm: and therfore his zeale in breaking the Tables, and indignation against that Idolatry, became him the better.
Here "unbalanced zeal" seems to amount to immoderate religious enthusiasm.
From Samuel Clarke, "Englands Covenant Proved Lawful and Necessary Also at This Time Both by Scripture and Reason Together with Sundry Answers to the Usual Objections Made Against It" (1643):
This question is so fully, and largely discussed, and the thing proved to be lawfull both in case of Law, and Conscience, by Mr. Prin in his third part, and by diverse of our Di∣vines in their Answers to Dr. Ferne, that I shall referre the Reader to seeke for satisfaction there, where it may be had abundantly, if they come with unballanced Judgements and shut not their eyes against that light which shineth therein.
From Jerome Alexander, "A Breviate of a Sentence Given Against Jerome Alexander Esquire, an Utter Barrester of Lincolns-Inne, in the Court of Star-chamber, the 17th Day of November, in the Second Yeer of the Raign of our Soveraign Lord King Charls, of England, Scotland, France, and Ireland, &c. With Exceptions Taken to the Said Sentence, to Unfold the Iniquity Thereof" (1644):
... he is an Hermaphrodite, begotten betwixt Hawk and Buzzard, puts his sickle into every mans harvest, will be medling, what soever comes of it, albeit most times in the conclusion, he finds little thanks for his labour; makes work to have work, and rather than stand out, will spin his tow into a mighty length, make much ado about nothing, that he may be seen to have imployent, when God knowes those into whose society and businesse he thrusts himself, had rather have his roome than his Company, for this Swan though he hath milkie feathers, yet hath he a black Carcase: with Athalia you shall have him crying treason, treason, when himself is the greatest Traytor; and like a mad dog, you shall finde him as angry barking at the Moon which he cannot reach (at him thats above him) and doth but crosse his humour) as at that man whom he gets presently by the shins, and cannot withstand his fury, like the Raven you shall have him no longer to stay in the Ark, then he hath need of Noah, for his own ends [h]e will shake each man by the hand, and like wax will be pliable to every Print, the sparrows feather serves him as well as the Swans, without check he passeth over his own faults, but doth laugh at all others sinnes, and like a child seeing his visage in a glasse thinks it another babye; like thunder, when he makes the greatest clap, he lets all but a little stone, his discourse commonly ends in a lye: like a light vessell thats unballanced, he rises and fals with every wave, oderunt, quos metuit, first he hurts, and then he hates, and ever after he lookes upon them as guilty of that shame and sadnesse, which in the sin he hath contracted, and thus makes hatred an Apologie for wrong, and out of the narrownesse, incapacity, and Antipathy of his owne minde he fals to an under-valuing of persons, even to their non entity and existence, as things utterly unusefull, because he sees not what use himselfe can have of them, herein discovering as much absurdity in so peremptory a dislike as a blind man should do in wishing the sun put out, not considering that he himself receiveth benefit at the second hand from that very light, the beauty whereof he hath no acquaintance withall he will insult over a mans sufferings, and where he finds a Cowardly and faint resistance, will domineer like a pig in Pease-straw, and as a resty jade will then shew his tricks, when he findes his rider fearefull to put spurs to his sides; ...
In this astonishing, breathless rant (in which a single sentence goes on for pages), Alexander applies the same metaphor of an unbalanced vessel that Harsnett did to apassionate and inconstant person who "rises and fals with every wave."
The seventh-earliest match likewise seems to apply unballanced to a material thing—although not to anything maritime related. From a 1645 translation of "Pliny's Panegyricke: a Speech in Senate: Wherein Publike Thankes Are Presented to the Emperour Traian":
To breed up their children the rich are encouraged by great rewards and equall punishments, but the poore to take care for theirs have onely one reason, a good Prince. Now children borne for his service, unlesse with a munificent hand he cherish, feed, and embrace them, he hastens the sun-set of the Empire, the sun-set of the State; in vain a Prince neglecting the Communalty, like a head to a defective body, maintaines a bulke staggering with an unballanced weight.
And from Stephen Rich, "The Answer of Captain Stephen Rich Commander of the State Packet Barques and Post-Master of Dublin to a Scandalous Information of Evan Vaughan, late Post-Master of the Same City (1649):
7. That Captain Rich, did contract with the State to keep two able Barques for their service only, but did keep but one for fourteen months together, yet hath received his whole stipend for two, and that one Barque did make but one Voyage in the States service from the 30. of September 1648. to the eight of April 1649. And whereas the said Barques should at all times be in readinesse for the States service, and at no time unballanced, or laden and cumbred with Merchants goods, and yet whither he had one or two, they were alwayes unballasted of purpose to take in Merchants goods or Coales for his own best advantage.
To the seventh charge this Defendant saith, That he ever kept two sufficient Barques for the use of the State, save only in that time that both his said Barques were carryed away as aforesaid and then he pro∣vided two Barques for the said Service withall the expedition he could, but denyeth that the same were unballanced at any time when there was occasion to use them for the States service, and as to that part of the charge concerning taking in of Goods aboard, this Defendant humbly offereth and saith, that although it bring in no profit to the Owner, yet it was never denyed the Servant, that whilst the Barques did attend, and before the Packet came to hand the men might take in some Truncks or Goods, which perhaps might produce for their advantage, and incouragement, sometimes ten shillings, or more or lesser as occasion offered; ...
Thus, in the eight instances of "unballanced" that an EEBO search turns up, three instances that explicitly focus on unbalanced boats, and a fourth refers figuratively to "unballanced soules" that are subject to drowning and to the church as an Ark of safety. These examples certainly don't prove that unbalanced arose first in the context of cargo loads aboard ships, but they do offer fairly strong evidence that unbalanced loads on vessels were an early area of usage of the term unballanced.
Finally, I want to point out the following outlier occurrence of the word imballanced, which appears in Gervase Markham, The English Arcadia Alluding His Beginning from Sir Philip Sydneys Ending (1607):
The other in the Gallioon with no lesse but rather a more inauspitious hād of death ouer his incoūterers, by how much more neer he grew vnto the iudgemēts of the beholders, augmented both his owne rage, & their opinions touching the excellency of his rage; distributing such vnresistaable blowes, that his sword was seen sildom or neuer fall with his hand, but a body deuided from a soule fell dead at his foote; so that as it was most likely to be imagined the poore in-habitants of that vessell, growne to the desperate willfullnesse of absolute desperatenes (which is by death to shunne death) willfully ran the ship against the Rocks, whose armed brest of too hie proofe for so slender timber, split her in one instant into many thousand disioyned peeces; the suruiuing remnant more willingly offring vp their liues into the hands of the mercilesse Sea, whose mercy they had not tasted; then to the subiection of his sword, whose vigor in punishing their breathles companions bodies, in their floating witnessed. But neither did the daring of their dispaires (which was the vnexpected entrance to this euill, neither the danger it selfe (wherein the preciousnes of his deare life was imballanced) neither the inacquaintance of the soyle whereon he was ship-wrackt, neither his many wounds, the losse of Fortune, Hope, Honour, Wealth, or other expectati∣on breed in him either amazement, feare, or desistance from the continuance of that reuenge, which from the begīning he with so great vertue pursued; for being now left to struggle with the vnruly waues, whose imperious billowes (many times counter-checking his desires) gaue him a feeling remembrance of his mortall constitution, he gathered new life, by the hazard wherwith the old life stoode indangered: and swimming with such beautifull cōlinesse as Triton is feined to do, whē he vshers Neptune to Venus banquets, with his sworde in his hand, which often ensigne-like he flourished about his head, as who should say, Danger is but the hand-maid to Vertue, or as if he would haue chalenged moe perils then those, imputing the escape of this no worthie reputation; without turning his eie backe vpon his owne safetie, he followed on still the ruine of those to whome was left no comfort but in ruinous perishing, and made such slaughter, that not a breathing soule was left to complaine, that so manie by one were become breathlesse; ...
I can't tell whether imballanced here means "unbalanced" or "put into the balance"—that is, figuratively, put at risk. In any event, the fact that the action described occurs on board a ship that is intentionally driven onto the rocks to escape death at the hands of a pirate seems merely coincidental with the "unbalanced vessel" instances of the 1630s and 1640s. It is also noteworthy this instance is one of only two that EEBO finds for the word imballanced/imbalanced from the period 1450–1700, the other one being from 1661 and involving "a graine imbalanced against the round world" and therefore "found light and to bear no scale at all"—a context indicating that the word there means "placed in balance against."