The most common and most likely answer to this crossword clue is the 6 letter word SUPPED
I've seen this crossword clue in The Daily Telegraph
S
U
P
P
E
D
More clues leading to the result SUPPED
Buddy explains!

The straight part
We know that "drunk" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case drunk = supped.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does SUPPED mean?
SUPPED verb- To sip; to take a small amount of food or drink into the mouth, especially with a spoon.
- To take supper.