Well the Fine Dictionary says it comes from Latin through French, quoting Chambers's Twentieth Century Dictionary:
Fr.,—L. finitus (finished), from finīre (to finish), finis (an end).
I tried to look up the word in the dictionary itself and it says:
ETYMOLOGY: 13c: from French fin (end), in the sense of 'boundary or limit'.
In his book, Modern System of the Arts: A Study in the History of Aesthetics, Paul Oskar Kristeller agrees with the time of origin of the term, and uses it as an equivalent of Beaux Arts:
At least a few scholars have noticed that the term "Art,"
with a capital A and in its modern sense, and the related term "Fine
Arts" (Beaux Arts) originated in all probability in the eighteenth
century. (the names of these scholars are: L. Venturi, R. G. Collingwoo, Parker and McMahon)
So the term fine arts is relatively new, being formed of already existing words"fine" and "art". You only need to look at the etymology of each word to know how they came to be. The problem is that fine has more than one meaning:
Here is what Etymoline has to say about FINE:
mid-13c., "unblemished, refined, pure, free of impurities," also "of high quality, choice," from Old French fin "perfected, of highest quality" (12c.), a back-formation from finire or else from Latin finis "that which divides, a boundary, limit, border, end" (see finish (v.)). The English word is from c. 1300 as "rich, valuable, costly;" also in a moral sense "true, genuine; faithful, constant." From late 14c. as "expertly fashioned, well or skillfully made," also, of cloth, "delicately wrought."
In French, the main meaning remains "delicate, intricately skillful;" in English since c. 1300 fine has been also a general broad expression of admiration or approval, the equivalent of French beau (as in fine arts, "those which appeal to the mind and the imagination," 1767, translating French beaux-arts). Related: Finer; finest. Fine print is from 1861 as "type small and close-set;" by 1934 in the extended sense "qualifications and limitations of a deal."
So Etymoline does record that it can mean that which divides, a boundary, limit, border, end but assumes that it was used in fine arts to mean which appeal to the mind and the imagination.
There is no unanimous agreement among scholars about this issue, so we can only look at it and accept that there may be more than one possibility to interpret this term.
This art site states that:
scholars agree that the term fine art equals to the “art for art’s sake”.
This would point out in the direction of finite art as an end in itself.
However, the same site explains that:
The French philosopher Charles Batteux was among the first to specify fine artwork in his work entitled Les Beaux-Arts réduits à un même Principe ("The Fine Arts Reduced to a Single Principle") in 1746. In this text, the use of the word “fine” with “art” is derived from the French “Beaux Arts”, simply meaning delicate, excellent, and finely made, art based on the idea of beauty and good sense.
In a way, the two possible meanings are not that contradictory: anyone would agree that fine art refers to refined, pure artistic aesthetics as an end in itself:
Final art is a visual art considered to have been created primarily for aesthetic purposes and judged for its beauty and meaningfulness (dictionary.com)
Definitions can be slippery, but in basic terms, fine art is something created for aesthetic or intellectual value rather than utilitarian or practical purpose.
We must agree, however, that the case for fine meaning beautiful is very strong, as it seems to be the established English translation of Beaux Arts:
In antiquity the fine arts were not explicitly named, nor even distinctly recognized, as a separate class. In other modern languages besides English they are called by the equivalent name of the beautiful arts (belle arti, beaux arts, schone Kiinste). The fine or beautiful arts then, it is usually said, are those among the arts of man which minister, not primarily to his material necessities or conveniences, but to his love of beauty. (Th.com.encyclopedia)
To finish on a humorous note, here is a "post-modern" etymology of the term :)