Dreadful bore's serious... (5)
I believe the answer is:
sober
'serious' is the definition.
(sober can mean sensible or serious)
'dreadful bore's' is the wordplay.
'dreadful' indicates an anagram.
'bores' anagrammed gives 'SOBER'.
(Other definitions for sober that I've seen before include "Staid; clear-headed" , "Serious; muted in colour" , "Serious or solemn and not drunk" , "Unintoxicated" , "Not intoxicated" .)