Plus, how would they define it drunkenness? I know it when I see it? Show of hands? Who, in the Constitutional rules, are “they”?
One of the folks with links to Senate Rules will have to answer substansively. Bt I can furnish one part – by the Constitution, each house of Congress is the sole judge of its own rules. So “they” are the members of the Senate, by I think a 2/3 or 3/5 vote (which means the V.P.'s casting vote doesn’t count – it would have to be 67 or 60 Senators voting to change the rules.)
Rules? They ain’t got no stinkeein’ rules. 
Ask Ted Kennedy.
Oh, wait…
No rules against it. Not following the rules (being too “out of it” to be orderly) will get you removed from the floor, but otherwise, you’re immune from most local laws while on the floor.
I’d say that being an open drunkard will make it hard to get re-elected, but history is not showing overwhelming evidence backing that theory.