The most common and most likely answer to this crossword clue is the 7 plus 7 letter word CAPITAL OFFENCE
I've seen this crossword clue in The Times
C
A
P
I
T
A
L
O
F
F
E
N
C
E
More clues leading to the result CAPITAL OFFENCE
Buddy explains!
The straight part
We know that "Serious crime" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case serious crime = capital offence.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.