Should I contact accomplished researchers listed as editors of shady journal? I recently got an unsolicited email advertising a questionable-looking mathematics journal.  Upon investigation, it looks pretty shady: page charges, manuscripts to be submitted in Microsoft Word (legitimate math journals always use LaTeX), overbroad scope, and a promise for peer review within two months (unreasonably short for mathematics).  

In its three years of operation, one particular author has been published in it three times, including an “elementary” proof of Fermat’s Last Theorem, and another paper that appears to be a proof of a statement for which an explicit numerical counterexample is known.  (Interestingly, it appears the same author has also published a proof of the Goldbach conjecture in a journal with a nearly identical name from another publisher indexed in MathSciNet, and has also apparently settled the Twin Primes and Collatz conjectures!)

(Since the preceding paragraph may not make any sense to non-mathematicians, let me say: this is roughly the equivalent of a physics journal publishing a paper that claims to have achieved time travel with household materials.  I should note that the journal has a subscription fee (which I have no intention of paying), so I can't actually read the articles in question; but their abstracts are pretty damning.)

However, the journal’s editorial board includes some names from reputable institutions; people with many publications in high-quality journals.  (There are many other names from institutions I know nothing about.)  Giving them the benefit of the doubt, it’s entirely possible that they are not paying attention to what the journal is doing, or they agreed to be editors without checking on the journal, or even that they have been listed without their knowledge (this has been known to happen).

Is it appropriate to try to inform these editors what’s happening in their names?  If so, how can I do it tactfully?

On the one hand, if someone was using my name on a shady journal, I’d want to know.  On the other hand, I don’t want to offend or embarrass them by just sending an email saying: “this journal you edit is crap”.  It’s even possible that they somehow approve of the journal (e.g. they have a philosophy that the world generally needs more journals and fewer barriers to publication), in which case, I fear no good can come of me criticizing it to them.
I wouldn't worry too much about tact.  There's enough information in your question to identify the journal, and it looks impressively bad, even by the dismal standards of junk journals.  It's so terrible that I'd consider it unethical to be actively involved as an editor, and humiliating to be passively involved.  Bringing this to the editors' attention would be doing them a favor: if their names are being used without their knowledge, then they'll find out, if they haven't been paying close enough attention, then they'll get a valuable wake-up call, and if they are already well aware of how terrible the journal is, then at least they'll learn that someone has noticed their involvement and disapproves.  Of course it would be quite an awkward e-mail.  Perhaps the easiest solution is an anonymous e-mail: it won't make things less painful for the editors, but at least it won't affect your relationship with them.