The most common and most likely answer to this crossword clue is the 14 letter word DRILL-SERGEANT
I've seen this crossword clue in Independent
D
R
I
L
L
-
S
E
R
G
E
A
N
T
Buddy explains!
The straight part
We know that "One responsible for exercising" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case one responsible for exercising = drill-sergeant.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.