A few necessities. Public toilets and WiFi - major hotel lobbies usually have both free WiFi and clean restrooms that you can use. As a Tour Guide, I use them all the time!
WiFi - Most NYC parks and all NYC libraries have free WiFi. Of course there also many cafés in the city as well, including McDonald's! And many of the subway stations also have free WiFi access now.
Public Toilets - This is something that New York City really needs to find a better solution to! Bryant Park has a very clean facility. Other places include Grand Central Terminal, Madison Square Park, fast food restaurants and larger stores like Macy's.