While technically a private institution, NYPL is chartered by New York State, many of its buildings are publicly owned, and the mayor, council speaker, and comptroller each appoint a representative to the board.
The NYPL board has discussed renovating and re-purposing these apartments. It's a good idea, though it should be pointed out that this board has a history of foolish, reckless, and possibly corrupt dealing, as when they sold off the Donnell Branch and nearly destroyed the iconic Main Library on 42nd & 5th (the one with the lions; look up "Central Library Plan" for details).
Now we are working on a local designation. This is important because real estate developers have many of these buildings in the crosshairs. Currently, hand in hand with the NYPL and the Robin Hood Foundation, they are looking to tear down the Inwood Branch, under the guise of building a shiny new library, inside a big condo with some "affordable housing". A few years ago, they did tear down the Brooklyn Heights Branch, replacing it with a smaller library inside a large condo (there is an argument that civic works ought to be freestanding to communicate import). Before that, the Donnell Library was torn down, replaced year later with a library that is a fraction of the original and squeezed into a basement beneath a luxury hotel, the latter valued at a multiple of what the NYPL got in the deal. And for a while, the developers behind the Barclays Center were itching to tear down the Pacific Branch in Brooklyn.
These libraries are important civic spaces. Whatever your feelings about paper vs digital, there is no denying that NYC's libraries are heavily used, and that they constitute an important part of the fabric of our civic life, a place where New Yorkers of all stripes congregate and mingle.
Historical footnote: NYPL is in fact a private institution chartered by New York State, formed through the merger of the Astor and Lenox libraries, to which was added a gift from the Tilden Foundation. It serves the boroughs of Manhattan, the Bronx, and Staten Island, while the Queens Public Library and Brooklyn Public Library have their own systems (Brooklyn was a separate city until 1898). To further confuse matters, many of the libraries NYPL runs, including the Main Library, are owned by NYC, while others are belong to NYPL itself. Scott Sherman's book, Patience & Fortitude, provides a good, pithy history.
Always felt YT sold itself early because they needed someone with deep pockets to defend themselves from the lawsuits over copyright infringement. Insiders, am I wrong?