HEre's an interesting notion as to derivation: buying rounds for the boys at the bar; presumably a rounder would be one who accepts bought rounds bought does not buy for others.

"Boys, when I've had money, I've often been found
Buying drinks for the house full and passing them round
And some that I've bought for, and treated them well
When my money's all gone, have consigned me to hell

There's them now that loves me and calls me a friend
So when I've got money, I'll be drinking with them
And all of you rounders can all go to hell
If my friends won't drink with me, well, I'll drink by myself
"