Trademark law works a little different than most people understand. Having a trademark on a certain word or phrase doesn't mean other people can't use that word or phrase, just that they cannot use it within the same field (the same trademark identifier can be used by different owners in different types of business) in order to draw the business of people expecting to find the other business.
The trademark in question is Lizzie Borden Museum (and not just Lizzie Borden), and I think the owners of the Fall River Bad and Breakfast have a very strong case. A new place in Salem can't just take the same name and use it to promote their own business. Both businesses are museums, both are in the same state, people will definitely be confused. The ones in Salem can come up with some other name that still gets the basic idea across. I think their best bet is to get some figures from Tassaud's or something similar and call it the Forty Whacks Museum...
Regarding Jack the Ripper trademarks, in the US there have been at least 8 filed, but only one is considered still active: Jack the Ripper Tableskirting (though it seems from other references that they primarily go by just JTR Tableskirting these days).
Well, the Salem group is using it in connection with a museum, so any museum that says Lizzie Borden on it as a title -- on a sign, in a brochure, on a website address or as an email -- is going to conflict with a trademark on Lizzie Borden Museum, yeah.
If this group had decided to use Lizzie Borden as part of a title for, say, a lumber company or a pet store I'm sure the Fall River bed and breakfast owners wouldn't object.
Trademarks aren't just the names themselves, it's how the names are used.