I've hunted Stewart a number of times for pheasant in the fall and turkeys in the spring over the years, but like with many public place, it is a madhouse on the opening few days. I started going during the week and it was much nicer. It is a huge piece of property with diverse habitat, so you have many choices of where to hunt. On the plus side, the hunting areas are numbered and you can only park 1 (or 2 in some) car in each spot and you are restricted to that area. This is nice so that you have the area to yourself. On the down side, you have to spend time driving around looking for open spots and if its busy, it might take you an hour to find one.
I think that they do a lottery/wristband system for the start fo the deer season.

Here is a link for more info:

http://www.dec.ny.gov/lands/22544.html

I hope this helps. Good luck.