I crept about, caught by quiet woman being alert (10)
I believe the answer is:
perceptive
'alert' is the definition.
(thesaurus)
'i crept about caught by quiet woman' is the wordplay.
'about' indicates an anagram.
'caught by' means one lot of letters goes inside another.
'quiet' becomes 'p' (music symbol for 'piano').
'woman' becomes 'eve' (Eve is an example**).
'i'+'crept'='icrept'
'icrept' is an anagram of 'rcepti'.
'p'+'eve'='peve'
'rcepti' going within 'peve' is 'PERCEPTIVE'.
'being' is the link.
(Another definition for perceptive that I've seen is " Aware".)