Responsible and trustworthy? That's not on (6)
I believe the answer is:
liable
'responsible' is the definition.
(I've seen this before)
'trustworthy? that's not on' is the wordplay.
'trustworthy?' becomes 'reliable' ('reliable' can be a synonym of 'trustworthy').
'that's not' suggests deleting specific letters.
'on' becomes 're' ('on' can be mean 'with reference to').
'reliable' with 're' taken away is 'LIABLE'.
'and' acts as a link.
(Other definitions for liable that I've seen before include "Legally obligated" , "Likely" , "legally obliged, held responsible" , "Exposed to risk" , "Subject to an obligation" .)