New York weather has its extremes and there will be a few periods of time when you may not be able to go out walking. Being close to the Atlantic, the area is subject to heavy storms and snow is fairly common. Hot, humid summers can cause problems as well, but walking early can help you avoid these issues.
New Dorp is flat which is good news for those that have not done a lot of walking or are just not interested in heavy workouts. The area has a real neighborhood feel to it, rather than a heavily urban atmosphere. There isn't a sidewalk on every street, but there are a fair number of grassy shoulders that will make your pooch happy on their daily walks. Many of the streets also have large shade trees which will help on the warmer days, but you must still use caution. Grassy areas and trees are not consistent through the neighborhood and if the temperature is high, then you can be fairly certain that the concrete will be hot as well. If it gets too hot, it can burn your dog's feet. You are better off finding a greenway or doing your walks early in the morning on the hottest days of the year.
Miller Field to the northeast of New Dorp is also a possibility. Once an Army Air Corps Base, it is now a 187-acre recreational area. Nearly all grass, this makes a great place to take your pup for a place to run and play.