It is the Bellevue Psychiatric Hospital Building (formerly the Bellevue Psychopathic Hospital when it was established in the 1930s). However, all psychiatric healthcare operations ended when this building closed as a hospital in the 1980s. All psychiatric care is performed on the main Bellevue campus, so "Old" is a bit of a choice on my part -- but it is an accurate descriptor.
I believe it's used as a homeless shelter today.
If you think it should be changed from being labeled as a "hospital building", in spite of its history as such, let me know what it ought be changed to.