The most common and most likely answer to this crossword clue is the 6 letter word DEMEAN
I've seen this crossword clue in The Guardian
D
E
M
E
A
N
More clues leading to the result DEMEAN
- . Academic head maintains setter’s lower status
- Humble churchman shows me inside
- Lower status of French ace
- . Head of faculty’s written about yours truly getting lower
- Put down setter, entertained by actor
- Clergyman keeps setter humble
- Humble cleric taking me to heart
- Cathedral dignitary, this person being held in dishonour
- Humble cleric covering the Middle East
- Humble cleric takes me in
Buddy explains!

The straight part
We know that "Put down" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case put down = demean.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 DEMEAN mean?
DEMEAN verb- To debase; to lower; to degrade.
- To humble, humble oneself; to humiliate.
- To mortify.
- Management; treatment.
- Behavior; conduct; bearing; demeanor.
- To manage; to conduct; to treat.
- (now rare) To conduct; to behave; to comport; followed by the reflexive pronoun.
- Demesne.
- Resources; means.
- To subtract the mean from (a value, or every observation in a dataset).