The most common and most likely answer to this crossword clue is the 10 letter word ABSTAINERS
I've seen this crossword clue in The Guardian
A
B
S
T
A
I
N
E
R
S
Buddy explains!
The straight part
We know that "They don’t put crosses on papers" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case they don’t put crosses on papers = abstainers.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.