Yes, I do think we should strive to get questions answered and accepted as such. This of course starts with getting answers, but where questions are too broad and can not be answered we as community should aid to get questions specified or flagged/voted to be closed.
Though some questions may also be hard to answer without specific knowledge which may not yet be available within this site's community, if the questions are well asked, it should be ok for them to remain open and unanswered/unaccepted.
When there are good quality answers to a question, but none is accepted as such, we could comment on the question to remind the poster to select one, but if this triggers no response I think no other action can be taken.