Unless we start talking criminal background checks, you are no safer than before. If anything it gives a false sense of security, which might make you less likely pursue further avenues of safeguarding. Of course criminal background checks are a completely stupid idea. I'm sure many good natured people would be hesitant to put this level of info in the hands of someone they don't know.
The truth is though even criminals can get credit cards, debit cards and any other method of verification you can think of. So that info doesn't really help. Worse still they can use fraudulent information just as easily. What do you do now? You also have to remember those opportunistic types, you know the person who has never really done anything wrong, but for whatever reason at that moment they are compelled to break the law or behave in a manner they normally wouldn't.
In a society that is mostly good, we have the police the courts, the law and prisons, to deal with bad people. You can not really on airbnb to provide you with a definitive solution to this problem, if they did then they should be running the country. Nor should you expect them to put systems in place that violate the rights of the majority, or systems that provide a false sense of security. The best they can ever do is provide you with support and advice on preventative measures.
Your safety should still be your top concern. You should not be relying on another to provide it for you.