Mortify a Bachelor of Arts, quiet (5)
I believe the answer is:
abash
'mortify' is the definition.
(thesaurus)
'a bachelor of arts quiet' is the wordplay.
'bachelor of arts' becomes 'ba' ('BA' can be a synonym of 'bachelor of arts').
'quiet' becomes 'sh'.
'a'+'ba'+'sh'='ABASH'
(Other definitions for abash that I've seen before include "Cause to feel ashamed" , "put one to shame" , "Strike with embarrassment" , "Crush" , "show up" .)