Keep in mind that most of the really important books, especially reference books, have never been allowed to be checked out from the library.
Chicago did this same policy about 2 years ago. It has similar demographics as New York, so you would assume that any problems New York is going to face would have already been seen in Chicago. And as far as I’ve heard, the benefits have far, far outweighed the problems. My family never had problems with late fees at the library (maybe $5 per year?), but the new policy definitely changed our library experience for the better.
Also, New York, Chicago, and many other library systems have extremely strong philanthropic organizations attached to them. It is very prestigious to be on the board and it is very prestigious to raise or donate money to the library. They are very well run and very well funded. Losing/Replacing 1-2% more books per year is not a financial issue for them. The ridiculous cost of eBooks are more of an issue than lost paper books.