If you want the best combination of price, HOTEL, being very close to Manhattan, but close to interesting parts of kool and "en vogue" Brooklyn, while still taking the train to the National Tennis Center (F to 7 train), stay at the Brooklyn Marriott. I think the Offical name is Brooklyn Bridge Marriot, or some sorts.
Needless to say, there a kazillion nice hotels to stay at in the city, all depending on your your price, style, planned activities and vicinity/proximity. But that hotel is at a nice location-walk able to the Brooklyn Bridge, should be cheaper than comparably priced hotels in Manhattan, and has a reputation for excellent service (especially if you plan to be there a week or two).
The Hotels around Times Square, Grand Central, Lexington, Park Ave Ave-near the 50's, and anything close to the E, F and 7 Train, will get you there quite easily.
Only you know what type of hotel you want to stay at and close to where. So call in advance, or make the best deal, then book it.
Just check some of the Zagats, Tripadvisor, Yelp, City Search, etc. customer and editorial reviews as you do your search to help.
Some of the advice you got previously received on the 1st week vs the 2nd week of the tournament, was good advice.