My husband wants to change his career paths but I don't think it's a good idea because we have kids and our home to take care of. He's really unhappy with his job but I think he should be more responsible. Should I prioritize his happiness and support him or tell him to do the responsible thing?

Sis, I think, it's better to give him a chance to look for other job / career. But make sure he'll only leave his present work if he gets a new one already. Just for security. Letting him stay at the job that he doesn't want will affect his performance. It might lead to demotion or wven worst, dismissal.