There's plenty they could do. They could offer insurance for this sort of situation (yes, they'd have to add an insurance fee or increase their rates for this to make business sense). They could implement credit card address verification, and require that a credit card be in their system for some period of time (3 or 5 days) with a hold on it before someone can make their first rental, which would make it a lot less likely that someone could effectively use a stolen credit card. They could verify phone numbers of guests and hosts by calling them (perhaps even with an automated system, that calls someone up and asks them to punch in a number they got when signing up for Airbnb), which would give an additional piece of evidence about someone's identity if needed. They could implement a trust system in which people with established histories using the system can vouch for someone else, and if no one vouches for you, then you need to do one of the above steps or some other form of additional identity verification. I'm sure there are other ideas that I haven't come up with that would also help and not be too cumbersome; these are just a few off the top of my head.

I don't think that any of these would make the service unusable; if they did all of them, it might be a bit cumbersome, but implementing one or two of these measures would be fine. They would add a little bit of friction, for sure, but I think that it would be worth it for avoiding these sorts of awful situations.