+1 on the helicopter suggestion, if you can do it.
Be aware that NYPD may roust you if you use a tripod. They're not supposed to, but some of them do and it really isn't worth arguing with them because the officer might decide to confiscate your stuff and you. I would suggest printing this out and carrying it with you to (politely) show to an officer if asked if you intend to use a tripod: http://www.nyc.gov/html/film/html/permits/permit_required_fee.shtml
- but if an officer tells you to move, move.
The empire state building do not allow monopods, tripods, etc. They go so far as to confiscate the tiny gorillapods for point and shoots at security.
On the Brooklyn side of the brooklyn bridge are two very popular spots - to the right of the bridge you have the dumbo area, and on the left side of the bridge you have the brooklyn bridge park (brooklyn bridge park is often more crowded, particularly at the end of the day). Go there for the blue period before dawn to have the buildings lit (sun behind you) or at night to try catch some sunset sky behind the skyline.
You can also take a ferry (or other transport) over to Liberty State Park in NJ for a view from the other side - sunset behind you hitting the buildings. The park opens at 6am, so you'll miss the pre dawn blue period unfortunately.
For midtown, you can try Roosevelt Island. If you're quick and lucky, you might be able to get a memorable shot or two from the tram on the ride over from the 59th street bridge stop. A shot of the U.N. building can be had from the southern tip of Roosevelt Island as well.
Don't discount your hotel room either. If it's a big building, see if you can get a room very high up. Turn off the lights in the room and you might have a great view.
If you take the staten island ferry, immediately go to the highest level on the right hand side. The statue of liberty will be on that side. You can go right on the way back as well if you want a shot of the Verrazanno bridge, a rather boring bit of brooklyn, and governor's island.