Women's Institute’s modern backing is common sense (6)
I believe the answer is:
wisdom
'is common sense' is the definition.
(I've seen this before)
'women's institute's modern backing' is the wordplay.
'women's institute's' becomes 'wi's'.
'modern' becomes 'mod' (synonyms).
'backing' says the letters should be written in reverse.
'mod' reversed gives 'dom'.
'wis'+'dom'='WISDOM'
(Other definitions for wisdom that I've seen before include "Prudence - insight" , "See 13" , "Knowledge gained through experience" , "Combination of experience, knowledge and good judgement" , "English actor, comedian and singer-songwriter (1915-2010)" .)