When I was younger I always thought only old people lived in New Bedford, primarily because my grandparents lived there. It really isn't the case. There are tons to do in the area.
The Ocean Explorium is a new addition to the area. Two years ago the Explorium opened in an old bank smack dab in the middle of the city, within walking distance from the port. The size makes it deceptive to how long you might stay. Though small, there is a shark and ray petting tank, Boog's favorite by far, a cold water local touch tank with crabs and sea stars, The Sphere which is a mostly enclosed mini theater area on which you can watch different Earthly presentations, arts and crafts, and a toddler exploring room. All in all they had a great visit and we stayed about 2 hours, just enough for the street parking meter.
Buttonwoods Park Zoo is another great place to take the kids. It has over 250 animals, 30 exhibits including elephants, bison, turtles, farm collection, seal and otters and a heck of a lot more. Another plus, it is fully accessible for those using adaptive equipment. Give yourself around 2 hours.
The New Bedford Whaling Museum This 107 year old museum is an educational tribute to the area's vast history of fishing and whaling. It includes the world's largest 1/2 scale ship model, whale skeletons, and it even has a kid friendly Whaling Adventure link on their site to get the little ones excited.
Lizzie Borden Museum If you have older ones, you may want to bring them to The Lizzie Borden Bed and Breakfast Museum and determine for yourselves if she committed the 40 whacks. Remember, this is an active ghost hunting house so be prepared....
Battleship Cove Is the world's largest naval ship museum. See the USS Massachusetts, PT Boats, Cobra Attack Helicopter, and a lot more. You can even do overnights there! There is a link to a printable coupon on its site (but I'll connect you here)
Fairhaven Bike path Just a few exits east of Downtown New Bedford is the bike path. It's about 3.5 miles long but if you are down here anyway, bring the bikes and head out...
Cap'n Barnacle's Mini golf Just next to the bike path is a great 18 hole mini golf course with $5 rounds, and $2 replays.
Also, check out 2011 Free Fun Fridays by the Highland Street Foundation It has listing of Massachusetts freebie days for many of the above as well other things in the area!