The most common and most likely answer to this crossword clue is the 6 letter word SOILED
I've seen this crossword clue in The Guardian
S
O
I
L
E
D
Buddy explains!

The straight part
We know that "dirty" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case dirty = soiled.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 SOILED mean?
SOILED verb- To make dirty.
- To become dirty or soiled.
- To stain or mar, as with infamy or disgrace; to tarnish; to sully.
- To dirty one's clothing by accidentally defecating while clothed.
- To make invalid, to ruin.
- To enrich with soil or muck; to manure.
- To feed, as cattle or horses, in the barn or an enclosure, with fresh grass or green food cut for them, instead of sending them out to pasture; hence (such food having the effect of purging them), to purge by feeding on green food.
- Dirty