Only in rare cases would New York City be responsible for a backup in your home. You'd have to provide evidence that some event led to the backup, like a negligent sewer or water main repair in your street. Most of the time, you'll have to pay for your own sewer service or drain repair.
Homeowners in Staten Island, NY are responsible for both upper and lower lateral sewer lines. Upper lines are located within your property limits, and lower lines extend from your property to the sewer main in the street. These lines drain wastewater to the main city line. You're responsible for any breaks or blockages that happen in the lateral lines, and the City takes responsibility for the sewer main.