- Edited
If you can find the old holidays thread there was a lot of discussion on New York a few months ago. I went in 2019, initially mainly to go to the US Open tennis but it turned into a much larger trip across USA. For New York we stayed in an Air BnB in Brooklyn which was very cost effective as we had I think 17 days there. It did mean a lot of time spent on the subway though (about 45 mins to Times Square). If you only have a week then I'd probably suggest staying somewhere more central around Times Square. Others may be able to suggest specific hotels. Edit: As James has whilst I was typing.
If you want to climb up the Statue of Liberty I think you need to book in advance (that's what I did), but if not bothered about going to the island to see it close up you get a good view from the free Staten Island ferry that goes past it - also get a good view of the Manhattan skyline from the ferry too. Probably wouldn't have bothered with the ferry but got in touch with some distant relatives (Father was my Grandfather's cousin I think) I'd never met who live on Staten Island and they invited us to their house. Might not interest you and you may not have time but it was only when looking up things to do near New York that I realised Philadelphia was so close (<90 minute train) so did a day trip there basically to run up the Rocky Steps which was on my bucket list.
Happy to answer any questions on any other attractions you're interested in. If you find the old thread I think I posted pretty much the same as this.