I've done this a bunch of times and yes the hudson river is open but the east river is no longer availabe to land airplanes due to the Cirrus crash that occurred there some years ago.
Here's the breakdown:
Get the Terminal map and also take a quick peek at the helicopter map on skyvector.com. The heli map actually has the route names that you will fly should you need to get a clearance.
Depending on where you are coming from, the routes change on how you do the flight. I fly out of FRG on long island and take one of two routes - south along the shore at 500' under the floor of the Kennedy Bravo, climb to 1000 and cross the verezanno bridge, always staying on the right hand side of the river. Be careful for the storm of heli traffic, they usually stay at 500' when doing the tours but if they transition the area they will climb up to 1000. Head north along the hudson and announce on the common traffic freq 123.05 different landmarks - verezanno, the lady, empire state, madison sq garden, central park, george washington bridge. Passing the george washington bridge to the north you can make a u turn and head back south now along the jersey side of the river and repeat. If from the north, just reverse the ordeal. Don't do any turns, manvs. while in the corridor because sometimes it is packed. (the other route is throgs neck bridge, laguardia tower, Roosevelt Island north tip, central park, hudson river southbound, need clearances through the bravo and this is where the heli map comes in hand.)
Watch out for TFR's around sports stadiums, the briefer won't tell you about them. When the yankees are playing, you can't do the tour. When the mets are playing you can't come from the east over laguardia.
You can get a clearance to do this route at 1500 with laguardia tower. They have a dedicated freq. for heli and passing traffic not landing there.
You can PM me if you need any more info, I've done a bunch of these tours already and can delve into more detail if you wish. It is a really spectacular sight. Enjoy