Well, the Old West was before Prohibition, and the first tracked appearance of the phrase pre-dated our speakeasies, but there's no definitive answer that I know about. Some info on the variety of theories at
Word-A-Day here, including the jumpers from the 86th floor of the Empire State Building, the standard height of a door frame, the Military Code for AWOL, the number of ladle servings in an Army soup pot, the depth of a grave, and some others.
Personally, not matter where it came from originally, I'm looking forward to when we "86 43!"
(a current usage referring to getting rid of G.W.Bush)