The most common and most likely answer to this crossword clue is the 6 plus 3 letter word SAFETY PIN
I've seen this crossword clue in Financial Times
S
A
F
E
T
Y
P
I
N
More clues leading to the result SAFETY PIN
- It joins any pets, if playful
- Peter has tiny carved ‘P’ inside napkin holder
- Means of keeping things from falling apart sure to add to endless work for secretary
- I fasten folds round baby’s bottom softly
- Occasionally iffy panties torn? Fix them with this!
- Security code? I hardly ever see the point of it
- This should not be removed from fine pasty after cooking
- The point of this fastener isn’t discernible when it’s used
- Clasp and wheel bearing breaking after almost year
- One conceals the point of security
Buddy explains!

The straight part
We know that "I fasten folds round baby’s bottom softly" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case i fasten folds round baby’s bottom softly = safety pin.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.