Apparently a woman who rented out her apartment in London through Airbnb for New Year's Eve got it trashed and another case is referred in the same article where the hosts found that their guests had not only trashed their home, but also had an orgy there. It really sounds like a nightmare. But if 35 million guests, according to the article, stayed on Airbnb properties last year and there were only 540 cases of property damage costing more than 1000 dollars, it means this happens in fifteen cases out of a million. That's extremely rare. (But I think the Airbnb Host Guarantee that "provides protection for up to $1,000,000 in damages around the world" seems to be a pretty good insurance, even though there doesn't seem to be an orgy clause.)
As a comparison 0,8 percent - eight out of a thousand - of the households in Sweden report they were victims of burglary the year of 2014.
I guess a better way to do the comparison would be to see the risk per night. Airbnb expected 80 million nights booked last year, which would make 7 cases of "significant property damage" in a million nights. Whereas the risk of burglary in Sweden is 2 in a hundred thousand nights. So the risk of burglary is about three times as high. It could happen, and I certainly have a home insurance. But I don't worry too much about burglary either.