My first few days in NY have had beautiful weather, so I've done a lot of walking through Manhattan's open spaces. Here's some notes and photos.
Roosevelt Island is this wacky bit of Manhattan borough, an island in the East River. There's not much out there, condos and hospitals. The reason to visit is the open space, nice long walks on both sides of the island and great views of the Upper East Side. Also grass, parks, even barbecue grills for anyone's enjoyment. It will be particularly nice to visit in 2013 when the southern park is fully open. It looks like they are preserving the creepy hospital ruins there as part of the park, a nice touch of gothic drama in an otherwise bucolic place. (Roosevelt Island is also a popular video game setting; several GTA4 missions start at that hospital.) The Tramway is strongly recommended; it's a short ride but the view as you go sailing into midtown between the buildings is terrific.
The High Line is New York's most famous new park, a nicely landscaped mile or two of abandoned railroad track. Having a park up above the city streets makes all the difference, a nice quiet remove. They also did a stunning job on the design with many great places to sit and watch other people. There's too damn many people, actually, and the linear park design makes it hard to get any space. So it's more of a people watching place than an open area.