I guess my idea of a good tavern doesn't include children, but I think I know what you mean. Jacob Wirth's would be an excellent choice; its 19th-century atmosphere is fantastic, it has great beers, and the American menu is perfectly decent; I don't think much of their German food, however.
There are a bunch of similar Olde New England-type places near Fanueil Hall that would also serve, though you wouldn't find the food anything to rave about, merely passable: the Green Dragon, the downstairs bar at Durgin-Park, and the Marshall House.
The Bell in Hand isn't bad if you just turned old enough to drink and don't mind swilling your Bud Lite from a plastic cup.
Any of the places around the marketplace are child friendly - or they don't stay in business. In the marketplace, Ames Plow has been there since the beginning and isn't overwhelmed the way the downstairs at Durgin-Park can be. There really aren't any dive bars for adults left in the area. Durty Nelly's in Haymarket, for example, replaced an old dive.
I'm not a fan of the Bell-in-Hand except on a nice day when the windows are open. Purple Shamrock has a nicer ambience as a bar, but almost every place around there is a bar / restaurant, from the old ones like The Black Rose and Clarke's to the newer places.
I once had a very enjoyable beer or two in that pub adjacent to the Park Plaza hotel. I think it's called "O'Connor's". I had about an hour to kill while in the area and was pleasantly surprised with the atmosphere, and the appetizer of 'pub sausages' that went well with a couple draughts.
I don't remember seeing any kids but they did have enough tables where they might be officially a 'restaurant' and would allow them
Also downtown is that Irish place in the crescent by the courthouse heck if I can remember its name though.