Sailor takes exercise to become thinner (5)
I believe the answer is:
taper
'become thinner' is the definition.
(I've seen this before)
'sailor takes exercise' is the wordplay.
'sailor' becomes 'tar' (tar is a kind of sailor).
'takes' indicates putting letters inside.
'exercise' becomes 'pe'.
'tar' enclosing 'pe' is 'TAPER'.
'to' is the link.
(Other definitions for taper that I've seen before include "Slender candle" , "Narrow gradually as candle may" , "Gradually reduce - spill" , "Become thinner towards one end" , "Get thinner - lighter" .)