The most common and most likely answer to this crossword clue is the 5 letter word MESSY
I've seen this crossword clue in Financial Times
M
E
S
S
Y
More clues leading to the result MESSY
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 = messy.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 MESSY mean?
MESSY adjective- (of a place, situation, person, etc) In a disorderly state; chaotic; disorderly.
- (of a person) Prone to causing mess.
- (of a situation) Difficult or unpleasant to deal with.