The long answer is an attempt to get you to help yourself, because there are quite a few "What projection is this data in?" questions.
You need a bit of guesswork, a bit of geographic knowledge, and this website.
The steps I went through are: first find an event on your website with a fairly easily identifiable place, which is important if you don't know New York (which I don't). I then used Google maps to find the streets and intersections described in my chosen event - this gives me a good general location to start from.
Next, I went to spatialreference.org and searched for New York. It came up with quite a few, but I knew from my Googling that it was likely to be one of the Long Island ones. They're all pretty much the same, but use different datums, so I chose the most likely sounding one, NAD83 / New York Long Island (ftUS) as a best guess. Clicking on this link, takes you to a page with details of the projection, and an interactive map showing the bounds of the projection. By zooming in on this map, and panning it so the red marker sits roughly on the place I found with Google maps, I looked at the Output coordinates line above the map. The numbers I saw there matched very closely the numbers in the LINESTRING, so I concluded that was the right one.
The next job would be to reproject the data to some other projection and overlay it on a known map, and check there are no obvious errors, especially near the edges, or if there is a uniform shift, which might indicate a wrong projection.
Finally never underestimate the amount of error introduced by the people digitising the data in the first place; this strikes me as a "close is good enough" dataset. You will go mad trying to fix errors that you think are to do with reprojecting, but are in fact poor quality source data.
The short answer is (probably) EPSG:2263