There's a very good sports bar, Champions, in the ground floor of the Copley Marriott. Most of the people there are locals, and I guarantee you that it'll be packed and rocking. You don't even have to go outside to get there from the Sheraton -- follow the walkway to the Prudential Center to the Marriott/Copley Place, and take the escalator down 1 level.

I'd agree with avoiding the Fenway Park area.