Resent a drunk becoming serious (7)
I believe the answer is:
earnest
'serious' is the definition.
('earnest' can be similar in meaning to 'serious')
'resent a drunk' is the wordplay.
'drunk' indicates anagramming the letters.
'resent'+'a'='resenta'
'resenta' is an anagram of 'EARNEST'.
'becoming' is the link.
(Other definitions for earnest that I've seen before include "Sincere, solemn" , "Firm" , "Fixed" , "Serious-minded" , "Solemn; pledge" .)