D.B.
Will you be using a car or relying on mass transit? I don't recommend a car in Boston proper if you can avoid it. The streets are often narrow, and parking can be expensive. If there's a game, everything within a half mile of Fenway is at absurd rates.
Fenway Park is on the Green Line of the MBTA mass transit. There are 3 lines that fan out like fingers reaching westward from the Kenmore Square area. The B line goes along Commonwealth Ave., the C line goes down Beacon Street, and the D lines goes from nearby Copley station down Huntington Avenue. The stations are underground subways at Kenmore and east, and street level trolleys heading west.
Mass transit is easy in Boston - at least it will be after the snow is gone.
When my son's track team traveled to Boston for regional races, they always stayed at the Holiday Inn Brookline/Boston 1200 Beacon Street Brookline. There is a restaurant on the premises and many local family-friendly places within walking distance.
If you are a AAA member, you can get great info from their free travel service.
Another Mamapedia mom traveled to Boston a few months back and she stayed with her kids in the Charlestown area. I can get the info from her if you want (there are a lot in the area). She and I had extensive emails back and forth to brainstorm.
How old are your kids? The types of tourist stuff you will want to do will vary tremendously based on their ages and the length of time you will be here. Have yo identified other attractions? Science museum, Aquarium, Art museum, Children's Museum, Boston Common/Public Garden, Esplanade along the Charles River, the Freedom Trail (including Paul Revere's House, old cemeteries, Faneuil Hall & its museum are just a few ideas), waterfront, North End (Boston's "Little Italy"), Bunker Hill….the list just goes on and on.
I can't believe $500-700 a night is routine - not at all. Give me some additional info and I'd be happy to help more.