Tempest, as a native I wouldn't call the Battery or the Upper Bay "the Hudson River". Not till you got past the Battery itself. The traffic south of the Battery can be real fun, we used to watch the Governor's Island ferry (property of the USCG and therefore at the right hand of God) duke it out with the Staten Island ferry, the commercial traffic, the recreational traffic, and the real fun, when the "new" high speed commuter ferries started up they figured THEY were at the right hand of God because they had to maintain their schedules. Ah, nope, a coupla folks got some nasty surprises from their "elders" on that.
But it still can get busy out there.
Problem on the Hudson proper is not off Manhattan but way further up. Think West Point and all the twisty places where the river is subject to shifting winds, tight curves, constrained drafts. Now add the barge tows, and when they come around a corner if someone is towing (not pushing) the tow is going to swing and you may not be aware of which way it is coming, or where other traffic is going to go to avoid it and keep in deeper water.
Some years ago a sailboat made the mistake of anchoring overnight on the Hudson, not in a designated anchorage. IIRC they got run down and the owner killed. Of course there have been accidents from long tows in the Sound as well, but at least there's usually lots more manuevering room on the Sound.
And some of the stuff that sails upriver, like the Tropicana Banana freighter (literally) is bigger than you would expect, with a matching blind spot under the bow.
Then sometimes after heavy rains, you will find bits of houses and whole trees coming down, just like Mark Twain's tales on the Mississippi.
There IS plenty of room for all, but daylight and a radio
and some planning ahead might all be useful. And a camera.(G)
I think technically, the Hudson River ends someplace surprisingly high up like Kingston. South of that it is technically the Hudson Estuary, a flooded river where seawater comes up twice a day, tidally. And you can often see the tidal bore.