Welcome to bumps on the "sharing economy" road.
Edit: it appears property management companies are using public court records to screen this person, preventing them from renting. That's not an AirBnB problem, that's a fair housing legislation problem.
I foreclosed on my underwater town home. Should I need to live in my car because my credit report has one negative item on it in 16 years? (Rentals of course pull your credit)
Housing is a regulated industry for good reason. Property managers are free to use data sources such as in this case, but there needs to be an appeal process. Otherwise, we'll start to have a class of people who a defacto homeless due solely to historical data.