Airbnb basically saved my hide for my first week in my new home here. It is relatively popular to use Airbnb here as hotelling is rather scarce and that is further compounded by the cost; even hostels are in short supply. Many sometimes go month to month on Airbnb here and alleviating part of the accommodation crisis here (for both tourists and residents).
Of course, there are several other websites which also can achieve the same thing, i.e. matching people who want to live here with those willing to let out a room or entire apartment. Airbnb tends to veer towards tourists and rather short term rentals (especially rentals where one is absent due to vacation), whereas other websites are more for longer term arrangements (e.g. colocating / flatsharing for months). Airbnb particularly charges a fee which it collects, which probably adds to the ire of the NYC parties concerned in this disagreement, i.e. it is a "clear" front for bona fide second rate room rental, as it were. Had it been simply someone on Craigslist putting out they have a room available, someone responding and then their coming to a "gentlemen's agreement" to terms of engagement (including payment, if any), there would probably not be many eyelids batted. I bet that has already been happening for yonks; naturally, though, Craigslist doesn't earn a cent in that engagement.
I'm surprised how underhanded some people can be in renting out their squalid place for Airbnb rentals. In NYC I'm not sure how one combats bedbugs continuously (indeed, various chain hotels of high repute have been caught out with a bedbug infestation problem at least once, and I definitely check for the signs of them when staying in NYC or any known bedbug area). Over here, people may have simple places but they seem to take a lot more pride in their rented place. Similarly, it would pay one who is renting to be respectful of the place, know the rules of the area and leave the place in as good a condition as it was rented to them. You'd think a bit of decency isn't much to ask for and goes a long way.
Why do rents increase when Airbnb rentals become common? Unless you have people who are simply buying up whole blocks of apartments with no intention of living there and only renting them out through Airbnb, that would be a big issue. Same as someone who say has rented in a place for a long time, then decides not to live there and instead put it up on Airbnb and live somewhere else. That would be rather unfair and I can understand how that becomes a pressure. But someone who owns a place and lives there, who decides to let a spare room (or just the couch) or what not through Airbnb should not have a marked effect on rental prices.
As for the problems with noisy tenants etc. I don't think that can necessarily just be pinned on Airbnb. It would be like saying that violent video games has a marked effect on the propensity of youths growing up to murder, abuse or other sorts of delinquency. (You may have your own opinion on that, fair enough). The difference is that a hotel with proper licences for people to stay at the property would have more relative powers (under the law) to control those who breach acceptable boundaries or rules.
Finally, though - if it is the law then it is the law, and there's little you can do about it. If NYC has banned it then so be it, so the real issue is simply the speed and degree of enforcement and to what degree can Airbnb still "slip under the radar" or will a similar system simply manifest itself ('underground', if you will) through another means, e.g. Craigslist. What stiffer measures the city can adopt to enforce that law would be interesting to see.
NYC I would not think has any issues with a "tourist problem", i.e. the tourism industry is stable enough that banning Airbnb (if you will) and thus possibly keeping the range of affordable tourist accommodation at some arms' lengths will not have a severely detrimental effect overall to NYC because so many people will visit the place anyway and they will end up paying a premium because everyone else will. Therefore, the city is not concerned at all about the effect of tourism and is justly more concerned with protecting the local / resident accommodation market and affordable housing.