If one's mother is Jane and their father is Dave, and their name is John Smith, would they say any of the following?
I was born John Smith by Jane Doe and Dave Smith
I was born John Smith by Jane Doe to Dave Smith
What's the proper use of "born by.." and "born to.."? Men say that their wife bear them children. How does one mention in this structure the mother and father both?
While one can't be BORN by one's mother, all of us are BORNE by or of our mothers.
One could, however, say that one was sired or begot by their father and BORNE by their mother, both of whom are their progenitors.
Your examples:
1) I was born John Smith by Jane Doe and Dave Smith; 2) I was born John Smith by Jane Doe to Dave Smith – neither example is grammatically correct.
Correct possibilities:
• I was BORN John Smith (meaning having certain qualities or characteristics from the time of your birth, in this case, the name John Smith), TO or OF Jane Doe and Dave Smith.
• I, John Smith, was BORNE [meaning carried or given birth to] BY or OF my mother, Jane Doe.
sired transitive verb: 1a: father
borne past participle of bear
bear transitive verb: 2a: to give birth to.
begotten transitive verb: 1: to procreate as the father: sire
progenitor noun: 1a: an ancestor in the direct line: forefather; 2b: a biologically ancestral form
(linked definitions courtesy of Merriam-Webster online)
We are veering into horse breeding terminology. "The foal is out of a mare and by a stallion" (Emphasis Added) http://www.chronofhorse.com/forum/showthread.php?55849-Breeding-Terms-Used-Incorrectly
I made this an Answer instead of a comment only because I thought it important to alert the OP to the horse-breeding connotation of his wording. I thought "horse-breeding" the instant I saw the post -- and all my horse info comes from Dick Francis novels. Little Eva's answer is good.
At this late hour and advanced stage of the discourse, perhaps I should remain silent, yet...
Background
Dictionaries are by nature descriptive, not prescriptive. Lexicographers examine histories of usage to synthesize or 'distill' dictionary definitions from those uses...unless the lexicographer is a mere compiler of definitions from external sources (pre-existing dictionaries, etc.). It is essential to remember that dictionaries are meant to detail resulting senses (definitions) from past and present usage of a word or phrase, not to pre-determine the future usage--however frequently they (dictionaries) may be used in a prescriptive manner, and however often they are cited as if they were some sort of final authority.
The Present Case
The first of your examples,
I was born John Smith by Jane Doe and Dave Smith.
is correct according to my sense of usage that will be readily understood and, at the same time, usage that will not foster (in me) a niggling sense of incorrectness. The second of your examples does not meet my sense of correctness. It is worth noting that my sense of usage is steeped in literature, rather than conversation.
If I look to a dictionary to confirm or deny my sense of the correct usage, I find my sense of correctness born out. For example, in The Free Dictionary, we find this:
bear 1 (bâr) v. bore (bôr), borne (bôrn) or born (bôrn), bear·ing, bears v.tr.
....
Usage Note: Thanks to the vagaries of English spelling, bear has two past participles: born and borne. Traditionally, born is used only in passive constructions referring to birth: I was born in Chicago. For all other uses, including active constructions referring to birth, borne is the standard form: She has borne both her children at home. I have borne his insolence with the patience of a saint.
and, later, this:
usage: Since the latter part of the 18th century, a distinction has been made between born and borne as past participles of the verb bear. borne is the past participle in all senses that do not refer to physical birth: The wheat fields have borne abundantly. Judges have always borne a burden of responsibility. borne is also the participle when the sense is “to bring forth (young)” and the focus is on the mother rather than on the child. In such cases, borne is preceded by a form of have or followed by by: She had borne a son the previous year. Two children borne by her earlier were already grown. When the focus is on the offspring or on something brought forth as if by birth, born is the standard spelling, and it occurs in passive constructions and in adjective phrases: My friend was born in Ohio. No children have been born at the South Pole. Abraham Lincoln, born in Kentucky, grew up in Illinois.
Now, I am perhaps obviously invested in confirming, rather than denying, my sense of correctness. Thus, I reason that the focus of your example is on the child subject, rather than on the mother (and father), and so the second usage note confirms my sense of correctness. The first usage note also confirms my sense of correctness, but rather more trivially than the second (the example is passive and refers to birth).
bornandborne?