The most common and most likely answer to this crossword clue is the 7 letter word COUTURE
I've seen this crossword clue in The Daily Telegraph
C
O
U
T
U
R
E
More clues leading to the result COUTURE
Buddy explains!

The straight part
We know that "Dior’s business" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case dior’s business = couture.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 COUTURE mean?
COUTURE noun- The production of high-end, custom-made clothing