Crumbling basement walls and asbestos
Based on your description, your basement walls likely have no asbestos in them. But that's the least of your worries; a crumbling foundation is a major problem! This is not a DIY job unless you are have professional experience with structural engineering, foundation repair, and concrete pouring. Call in a pro. And make sure this doesn't happen again by improving the drainage outside of the basement. There are many ways to do this, from improving the grading to installing drain tile to building a subterranean roof. It all depends on the conditions at the site.