This comes up a lot; you want to take some photographs but you don't want to be treated like you might be a terrorist. It seems like headlines about photographers being hassled and even assaulted by law enforcement are perpetually flying. A BBC spoof site, NewsArse, even put out this satirical article called Terrorists ‘sick of being treated like photographers’.
If you are looking forward to photographing in a city, especially one that keeps close watch for terrorist activity like New York City does, the best thing you can do is know your rights.
Can you photograph in public in NYC? Yes, of course you can. Can you photograph in the subway system? Yes, it's explicitly permitted (as long as you are not interfering with the operation of the subway or blocking access, etc.). Do you need a permit to use a tripod on the sidewalk? No, you don't, but if you try to set up your tripod on a busy sidewalk, New Yorkers might kick you, so try not to be a nuisance about it.
Check out NYPD Operations Order 14, wherein the guidelines for field officers dealing with photographers are spelled out in detail:
I have read of photographers printing out similar documents to carry with them in the event that they are stopped, and that may not be a bad idea at all. Just remember a couple of really important things:
Know exactly what your rights are. Research the location you are going to be in and familiarize yourself with the local ordinances.
Be courteous and respectful. If an officer is giving you a hard time because they like it, being rude to them is only going to make matters worse. Be understanding of their position, and firm in stating your rights.
Comply with directives. If an officer asks you to do something, short of deleting your photographs (which they cannot do without a warrant, at a minimum), do it. You may state that you are complying under protest, but don't get into a situation where you are resisting, because that can lead to real charges against you.
Following these simple rules will ensure you have a productive and safe shoot!