Drunk gets thinner -- not one to beef up (10)
I believe the answer is:
strengthen
'beef up' is the definition.
(synonyms)
'drunk gets thinner not one' is the wordplay.
'drunk' indicates anagramming the letters.
'not' is a deletion indicator.
'one' becomes 'I' (Roman numeral).
'gets'+'thinner'='getsthinner'
'getsthinner' with 'i' removed is 'getsthnner'.
'getsthnner' anagrammed gives 'STRENGTHEN'.
'to' is the link.
(Other definitions for strengthen that I've seen before include "Reinforce, substantiate" , "Make tough" , "Make more powerful" , "Fortify" , "Brace" .)