They do. Without them the arts would be in even more of a sorry state than they already are. Compared to continental Europe the arts - like the railways - are relatively underfunded. Things are getting better, however, though sometimes it seems a lottery as to who gets cash and who doesn't.
Governments should get involved to the extent that money should be available for non-commercial arts projects, but not - unless it's a commission - have any say in what should be funded. That's what arts councils are for. And, yes, they do do some good, though sometimes the criteria for allocationg funds seems a little arbitrary.