hmmm. Groucho's bit was the cigar, not a drink. Henny Youngman was "take my wife, please." Jackie Gleason was a big drinker, but I don't remember that being part of his act. WC Fields? Didn't he have the red, bulbous nose of a drunk? There was another comic from that era, who was a physical comic and not so much about drinking. Buster Keaton, I think. I'll go with Gleason or Fields.
PBS had a six part special on the history of comedy called "Make Me Laugh" a few months ago. I think it's re-running now. Not too funny, but interesting.