Another way to look at it: Do you take better care of the car that you own, or the car that you rent?
I think the nature of housing in NYC makes it difficult for AirBnB places to function the same as they do in, say, Asheville NC. Renting a house is one thing, renting an apartment with a shared entrance and more permanent tenants attached to nearly every other wall is different in so many ways.
I am pretty careful.
I'd say the overwhelming majority of AirBnB hosts don't rent out their rooms to just anybody. They vet candidates to make sure they are respectful people with whom they themselves would get along and whom they think are going to be as respectful if not more respectful of their own property. At the end of the day, your long-term neighbors, people you apparently are fond of and trust have chosen to hand over the keys to their apartment to people they have decided to trust based on some criteria of their own choosing. It's perfectly natural to extend my trust with person A to person B, if I know person A trusts person B.
One of my friends is the number one AirBnB/Couchsurfing host AFAIK. Something like 700+ people have stayed with him between both sites over the past few years and I think he has had maybe 1-2 bad experiences, and those experiences did not involve the person being disrespectful of the building, property or neighbors, but just being people of a generally unpleasant attitude.
People treat hotel rooms shitty, but they generally treat an AirBnB rental with respect because they know that that is somebody's home and that someone trusted them with their home. The actual, implied or imagined presence of others does have a positive impact on behavior. Social influence, specifically compliance, most definitely is a factor at play, both in terms of the hosts expectations of the guest and the fact that each and every AirBnB guest has a reputation to upkeep in order to maintain good standing on the site.