If you need to be in Manhattan because of your job, but don't want to buy a co-op there because it's both very expensive and you won't find much green space, then you should look north of the city. There are lots of neighborhoods in both The Bronx and Westchester that provide you with lots of fresh air, green space, and also have a super easy commute to the city. Below you will find an overview of two really nice neighborhoods that you should look into when searching for a home that has access to lots of green space and fresh air.
Yonkers is a city in Westchester. However, it's really close to Manhattan. It's in the southern part of Westchester, right above The Bronx. There are trains that run into midtown Manhattan, so you can take mass transit to work. You can also drive in really easily, if you prefer to take your own car.
One of the cool things about Yonkers is that you can get either a co-op, condo, or a house. Some of the smaller towns in Westchester are mainly houses, so if you prefer an apartment over a house, then one of the fancy condo buildings in Yonkers is the way to go. The area also has a huge shopping center (Cross Country Square) where you will find everything you want. There is also a gigantic Costco and Stew Leonard grocery right off the highway, as well as a Whole Foods and medical center at the luxury Ridge Hill shopping center.
Even though it's a city, there are lots of trees, rivers, and lakes. You can take your family to Tibbets Park on the weekends and hang out in the shade of the beautiful oak trees. Or you could head over to the Hudson River promenade and walk alongside the water and have lunch at one of the cafes. The city has one really busy section (Broadway and north to Ghetty Square) but if you look around the Tuckahoe area you will find that it's very quiet and peaceful and mostly tree lined streets, small condos, and well maintained parks.
If you want to stay in New York City, but still want the feeling of living outside of the madness and congestion, then Riverdale is perfect. The neighborhood is in the Northwest and sits right on the Hudson River. It's actually south of Yonkers, so you could drive around both neighborhoods when looking at places. Riverdale has some classic, old, brick buildings, as well as huge high rise co-ops along Independence Ave and Riverdale Ave. If money is no object, then definitely check out the private homes west of the Henry Hudson parkway or the area known as Fieldston. These homes are beautiful and you get big backyards. There is even a lake in Fieldston, though its small and people don't swim in it.
One of the biggest green spaces in New York is right near Riverdale. This is Van Cortland park. There are horse stables where you can go riding (the horses go through the backwoods trails and you can get instructions to go with you), hiking, fishing at the Van Cortland lake, or golfing. It's a gigantic park that rivals Central Park and transforms the entire area into a really nice, fresh air space.