I concur with BTinNYC - if this is just for the purpose of getting around and seeing the city, then CITI Bike is the solution. You can pay as you go, and as long as you have a smartphone, plan ahead where you can both pick up and drop off a bike. It's like a bike-share platform in any European city. As far as actually renting a bike, any places that allow for this are going to be expensive and would only lend bikes for several hours. Frankly, I would avoid these services. CITI Bike is the way to go!
Also, as far as safety in the city, contrary to popular opinion (and my own personal thoughts), NYC can actually be a very safe and rewarding place to ride around - as long as one goes slow and uses bike lanes, looks around and pays attention to cars, you should be good. The fact that I myself almost die every time I go for a ride has to do with the neighborhoods I ride through (few bike lanes) and my riding style (no patience).