Yes its safe. You will be able to walk a block or so down to Rockefeller Center in the morning and stand outside the Today show and wave to all your friends back home and meet the celebs. Walk the other direction and see Times Square. You can hit all the pishy poshy shops on 5th Ave. Trump Tower and of course St. Patricks Cathedral. You are also right near Broadway and I stringly recommend catching a show. We will never forget our first Broadway Show. One night I walked alone down to the grocery and internet cafe and felt completely safe.
I had to send d/h to NYC on business between Thanksgiving and Christmas. He was teaching in the Empire State Building and you should have heard some of the horror stories the students had to tell about their NYC hotel rooms. That is a peak time so of course prices were up and selction was down. Many of the students found themselves paying well over $200 night for a twin bed and they had to stand on the bed to open a door or a drawer and shared a bath. So in my opinion Radio City Apts. is a steal, but do not expect the Ritz, it is clean, functional and affordable, not to mention spacious by NYC standards. I have stayed at the Marriott Marquis and they are very close in location, the Marriott is twice as much and we would not have had two bedrooms and kitchen, so given a choice between the two I would choose Radio City even if the Marquis was less expensive, because it suited our needs better. We stayed there pre and post cruise.
If you want to go to the Statue of Liberty (chic with the torch lol) you will need to go to Battery Park then catch the ferry out to tour her. We took the subway there and it wasn't hard to get there at all. It was cool too because they were filming a movie at the park. In all truthfulness though you'll have the best view of the Statue of Liberty when your ship leaves Manhattan. It is breathtaking. You will go right past her.