The most common and most likely answer to this crossword clue is the 3 plus 8 letter word EAR TRUMPETS
I've seen this crossword clue in The Guardian
E
A
R
T
R
U
M
P
E
T
S
Buddy explains!
The straight part
We know that "What you’d associate with Beethoven" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case what you’d associate with beethoven = ear trumpets.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.