I will be in NYC for the US Open the last week of August. I am looking for a nice hotel, queen beds nice size bath. There are just two of us traveling. Also want the hotel convenient to transportation to the open. What would be the best area to be in?
Would like to spend about $300.00 per night

I can't help you with a hotel, but transportation is easy to the open. The #7 train goes write there. Just about all manhattan subway lines (I think the only exception is the J which only runs to downtown anyway) stop either at Times Sq or Grand Central. The #7 can be picked up at either of those hubs.

The Long Island Railroad (from Penn Station) also goes to the US Open. Although I don't recommend staying near Penn STation (Most subway stop right there or very close a well).

Would Battery Park be a good location fou us? As far as catching the subway to the Open....

Most hotels in the Battery Park area (Embassy suites?) are not right near subway stops. Figure about a 10 minute walk. The you need to take a train to Times Square (10-20 minutes, depending on train line) and then take the #7 train to the US open. SO, it's doable, but not the most convenient. Most convenient wid be staying in Midtown manhattan, close to 42nd street (east or west, since the #7 starts at Times Square but also stops at 5th avenue and Lexington Avenue on the east side)
Midtown hotels will probably be quite a bit more expensive, though

As mentioned, the 2 options for getting to the open are the #7 train which runs along 42 st subway map here
http://www.mta.info/nyct/maps/submap.htm
or the special trains put on by the Long Island Railroad that leave from Penn station 34 st. & 7th ave. While you can certainly get to either from Battery Park, it wouldn't be the most convenient (refer back to the subway map).

The last stop in Manhattan for the 7 train is at Grand Central Terminal (42 st & Lexington) so any hotel near there would work well. Looking on quikbook.com there's a good rate for the Shelburne Murray Hill in a 1 bedroom suite (this is a nice Affinia hotel). There's also a good price for the Dylan ($259 plus tax), also near there. The Affinia Dumont is not far from Penn Station and not far from Grand Central so another good option.

The Grand Hyatt is literally attached to Grand Central but a bit corporate imo.

If you plan on doing any theater, you might prefer staying on the west side. The Millenium Broadway has a special on travelzoo.com

I was wondering about Battery Park, because of the Ritz Carlton being there. It seems that the feed back I am getting is to stay near 42St, take the subway. I want a nice Hotel, what are some 4**** it this area?

Try the Library Hotel. It is near Grand
Central Station. We loved the hotel.

The Library is a good option although I don't think it's considered 4* b/c it doesn't have a restaurant. The Dylan that I mentioned is very near, does have a restaurant and I think has larger rooms. They're including breakfast with room rate through Sept. 2 on their website but it's $289 while the Quikbook rate was $259.

The Bryant Park hotel is trendy and hip and on 41 st. across from Bryant Park. The Morgan and 70 Park are other places to look at. The Fitzpatrick Grand Central is right across from the terminal and many people like it but I think the others mentioned will be nicer.

There are a string of hotels on 44 st between 5th & 6th aves. that would also be close to the subway stop. They are the Royalton, the Algonquin, the Iroquois, City Club, the Sofitel, the Mansfield. The Sofitel is usually rated most preferred in the city on tripadvisor.

I was going to suggest Priceline, but if you "Would like to spend about $300.00 per night", then it would not be for you. They have good hotels at much lower rates.

Of course, first you have to look at biddingfortravel.com to learn how to bid.

I was thinking of the Grand Hyatt but it doesn't seem to have availability in your price range for that weekend.
In addition to others mentioned, the Roosevelt hotel is on 45th and madison but maybe not as fancy as what you are looking for. I think room sizes vary a lot.
You are going to find that a lot of hotels in mid town are more than $300/night unless you get a deal. Battery Park really would not be convenient for getting to the Open and the area doesn't have much night life.