I went camping 3 weeks ago up to Clarence Fahnstock park (which is our favorite north of the city camp). We do own a car, so we did not have to be worried about the rental. We HAVE gone up via the metro north to Cold Spring and called a cab to drive us the rest of the way, but that was when we were broke and didn't have a car yet
At Fahnstock: they regularly cleaned the bathrooms, everyone wore masks when not at their site. And I peed outside 50% of the time any ways. They also closed off any sites that were next to each other for added safety.
There were quite a few obvious first timers camping (a lot of people who only stayed 1 night) and didn't have much gear.
All that, it's not "risk-free" - it all hinges on how you feel about the rental car. But if you do research and feel good about that, all other aspects should be pretty low-risk once you are at the campground.
There also is always camping at Flloyd Bennet Field in BK. It's not as relaxing, but you can maybe get someone you trust to drive you there OR if you were feeling really badass, get your gear on bikes and ride there.