Our first suggestion is to not stay directly downtown; if you can stay out in the suburbs your hotel bill will drop by half for 20 minutes driving.
Club Quarters in Boston. Convenient to Government Center, Quincy Market, Beacon Hill, Downtown Crossing, and The Alley Bar.
Fenway Howard Johnson's. Backs onto Fenway Park, near Kenmore Square, Back Bay, Longwood Medical District, and across the street from the Boston Ramrod.
Chandler Inn Hotel. In the gay South End, next to Back Bay, walking distance to all of downtown, Fritz is it's hotel bar and The Eagle is down the street.
The Midtown Hotel. Across from the Christian Science Center & Symphony Hall, in the Back Bay next to the South End & Fenway, it's not directly adjacent to anything notably gay but it can be reasonably priced if nothing fancy.
Jurys Boston Hotel. In the heart of Back Bay next to the South End, located in the former Boston Police Headquarters, a stylish boutique hotel with a great restaurant and not far from a number of bars.