Boarded at foxhill for years with my two horses. They have turnout, (lots of barns don’t) and access to the well maintained Rockefeller trail system.Take MetroNorth from Grand Central to Pleasantville (about 40 min) then a cab (5mins) to the barn.
Back when I was there board required lessons, and no jumping without a trainer present (but not necessarily in a class).
Usually the horse had to be trained by staff as well. They were sensible and safety conscious and very experienced, barn is run by second generation family.
They don’t specialize in jumpers, rather hunter type, but you could call and see if they would accommodate. Don’t know their prices, I eventually couldn’t afford it.
Twin Lakes (Eastchester) offers board and lessons. Nobody uses the trails, but they exist. Your horse needs to be bike broke, I’ve ran into bicyclists in groups and motorcycles (illegal) when I boarded there. I was there under different management, so can’t speak to that.
P.S. congestion pricing on drivers, plus toll hikes make it harder. Two years ago I got a monthly parking in NYC for $600.