The most common and most likely answer to this crossword clue is the 6 plus 7 letter word EXCESS BAGGAGE
I've seen this crossword clue in The Times
E
X
C
E
S
S
B
A
G
G
A
G
E
More clues leading to the result EXCESS BAGGAGE
Buddy explains!

The straight part
We know that "costly at airport" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case costly at airport = excess baggage.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.