The most common and most likely answer to this crossword clue is the 8 letter word SEDATING
I've seen this crossword clue in The Guardian
S
E
D
A
T
I
N
G
More clues leading to the result SEDATING
Buddy explains!

The straight part
We know that "Making dopey" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case making dopey = sedating.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 SEDATING mean?
SEDATING verb- To calm or put (a person) to sleep using a sedative drug.
- To make tranquil.