On the other hand, I think fanfic promotes the franchise overall and helps keep interest and loyalty alive. Fans who band together and form creative communities like this one are never a bad thing for a franchise. That kind of loyalty and enthusiasm is a big deal. It wouldn't be tolerated if it wasn't beneficial on some level, IMO.
As someone who's published fanzines since 1979 and written for them since 1975, I can assure you that during the 70's, fanzines and fan fiction were quite beneficial to the franchise. The studio was well aware of it (how could they not be with Star Trek Lives!
extolling its virtues, and Star Trek: New Voyages
publishing fan fiction from fanzines?), and even used it to their advantage (during the 80's, fanzines would receive letters from Susan Sackett, Gene Roddenberry, Harve Bennett, Sylvia Rubenstein, etc., often to request we write and support this or that or the other thing).
In fact, I chose not to answer the poll because, to me, the choices of "beneficial to the franchise" or even "very beneficial to the franchise" aren't provided to select. In fact, the question itself is somewhat biased against fan fic.
And to answer the question, "Is it illegal?" Certainly. It's by the grace of the studio that they've never once shut me down for either the fanzines we produced or the fan films we're begining to produce. As my wife has pointed out time and again: it's good for the studio, otherwise we'd've been given C&Ds years ago.