Thanks Attilla and Budgie for the heads up, I hadn't seen this would be a problem at all until your posts.
I've now done some extensive research on the issue. Our host seems legit and the money has been sent and secured by AirBnB so if he turns out not to exist, we won't have lost the cash (but will be short of a place for a week) as they hold onto it for 24 hours after check in.
The rest of it looks complicated. I understand why NYC are minimising the potential for short term rentals and know about the current legal wrangles about handing over data to the authorities over there... terrible as it might sound, it looks, currently, that if our host isn't scamming us (i.e. does have an apartment to rent and is willing to do so for the price we have paid, and is worried enough about his AirBnB ratings, which he has a lot of, not to crap out on us) then out chances of being out on the street are slim. It also looks like (afaict) from internet searches that it's a building with only 4 units in it, one of which he lives in, so might not fall within the legal difficulties mentioned in the articles anyway.
The risk then is nice cheap apartment (it's really only fit for a couple, but you could shoehorn in other people, so the price looks right) with low risk of being evicted by the building supervisor vs expensive hotel with no risks of that kind, but with rubbish facilities.
Knowing the risks, I think we are willing to take it on the chin, especially having read a lot of testimonals of others who have been fine, and none from people who have fallen foul of it. More problematic is the moral element of people using AirBnB when they should be renting out the space to New Yorkers (the reasoning behind the bill) but harsh as it may sound, living in another global city (i.e. London) where all house prices have been driven out of affordability by a range of overseas unregulated buyers, feel that I've already taken the karmic sting on this one, and that our week in East Village is just not that big a deal in the long term.