How to assess the long-term stability of a college as part of a job search
While college failures are rare they do happen, especially in generally poor economic situations. But I think that mismanagement is more likely to be a factor when they do fail. But some advice, probably not complete. 

I hope the place is regionally accredited with one of the major agencies. Otherwise, not a great bet. 

I hope the place is not for profit. Otherwise...

I hope the place is a bit diversified in course offerings and student body and faculty and administration. A place traditionally founded and run by a single family is less of a sure bet. 

I hope the place draws students from a wider, rather than a narrower, geographical area. A national or international student body is best, but a few places can make do with only a regional reputation. How wide is their reputation? 

I hope they don't need to accept every applicant for financial reasons. Otherwise they have no cushion when times get bad. 

It is good if they have some endowment that they manage wisely as a cushion and to enable opportunities. 

I hope that they are open to new ideas, both in curriculum, and in management, but that is hard to judge. I hope that they don't need to chase every new trend to stay viable. 

I hope that the faculty is collegial, both within and across disciplines, though that is more of an indicator of a good work environment than longevity. 

I hope that a lot of decisions are made by the faculty and that the faculty is generally listened to in important discussions; curriculum and beyond. And that the administration respects those decisions. 

I hope that the board of directors is also diversified, not just cronies of the college president. 

I can probably think of more and will update as it occurs. I've worked at a couple of problematic places. None failed, but they did fail at being a satisfying place to work at times.