The answer depends a bit on what you mean by "with family"
Kids? Young kids or older?
Ideas:
Aquarium. It is relatively small, but I think well done. My 5 year old really likes it.
Rent bikes and take them from downtown Boston along the SW corridor (basically the greenway that runs along the Orange Line) to Franklin park and the zoo. If you do this, make sure to check out Doyles Cafe off Green Street in Jamaica Plain (
http://www.doylescafeboston.com/) . It is a fantastic, historical pub that is off the beaten path but has a lot of history (murals inside of former Boston mayors and politicians -- side story, my wife used to work for the City of Boston Parks Dept and they celebrated major events here, like promotions, etc).
Duck Boat tour of the city. It is touristy, and might seem pricey, but I think it is well worth it. You see a lot, quickly. Just make sure to plan ahead and get tickets (near Prudential Center).
Eat at the top of Prudential Center. There's a pricey restaurant (Top of the Hub) but also a bar area where you dont (or at least in the past didn't) need reservations - but get there early. Great views of the city.
See a game at Fenway Park. Yes, I know the Red Sox aren't any good this year, and I know this is Celtics blog, but Fenway is one of the oldest and is a special place, in my mind.
In terms of where to stay:
In Boston/ Cambridge/ etc its going to be pricey. You could look in Salem, though perhaps not many options (Hawthorne Hotel?) The nice thing about Salem is it is easy to take a the T (commuter train line) into Boston. There is also a ferry from Salem to Boston that I've never done but is quite popular with commuters. There's also good stuff in Salem, like the Peabody Essex Museum (really a very nice regional museum) and House of Seven Gables if that's your thing, but also more touristy stuff.