Graduate isn't worried about old supporter (7)
I believe the answer is:
bastion
'supporter' is the definition.
(I have seen 'staunch supporter' mean 'bastion' so perhaps 'supporter' could also mean 'bastion')
'graduate isn't worried about old' is the wordplay.
'graduate' becomes 'BA' (someone with a Bachelor of Arts degree).
'worried' indicates anagramming the letters (worry can mean to interfere with or disturb).
'about' indicates putting letters inside.
'old' becomes 'o' (common abbreviation eg in OE for Old English).
'isnt' with letters rearranged gives 'stin'.
'stin' going around 'o' is 'stion'.
'ba'+'stion'='BASTION'
(Other definitions for bastion that I've seen before include "Strong defender" , "Obtains in fortified place or stronghold" , "Projecting part of a fortification" , "A projecting fortification to increase angle of fire" , "Obtains (anag.)" .)