Serious whimper with tears regularly shed (5)
I believe the answer is:
sober
'serious' is the definition.
(sober can mean sensible or serious)
'whimper with tears regularly shed' is the wordplay.
'whimper' becomes 'sob' (both can mean to cry).
'with' means one lot of letters go next to another.
'regularly shed' means one should take alternating letters (letters shed at regular intervals).
The alternating letters of 'tears' are 'er'.
'sob'+'er'='SOBER'
(Other definitions for sober that I've seen before include "Sensible - clear-headed" , "Restrained, grave" , "Unintoxicated" , "Sedate and rational" , "Serious, thoughtful" .)