Following on from Bob's reference to McNeil and the 1853 date, McNeil goes into more detail in Southern Mountain Folksongs at p.154.
'This lively drinking lyric is a variant of the song Arthur Loesser includes in his 'Humor in American Song, pp.304-5. As Loesser indicates, the song is known both as 'Bingo' and 'Here's to good old Yale, drink it down, drink it down,' with the latter being the earlier title. This song dates from 1853 and probably was written by Julien Carle, about whom nothing else is known, but the lyrics usually heard today date from an 1861 arrangement by H. T.Bryant. Today the refrain of 'Balm of Gilead' is used commonly for the song, but Bryant featured a tune strongly reminiscent of 'The Old Grey Mare'. Neither of those melodies is used in the version here (Drink 'er down')----it could mean that the song was in existence prior to the 1853 Yale song.'

There may be further info in Loesser and an investigation of Yale songs of that period might throw up more info.