Close Encounters has its fans, but it doesn't have a fandom in the vein of Star Wars, Star Trek, or just about anything published under the Marvel label—and, yes, we're counting Squirrel Girl.
You won't find anybody cosplaying as Barry at the local sf convention, and while the film spawned a board game and pinball machine, it wasn't what you'd call an evergreen property. Fans don't go full-on "Thriller" and stage reenactments of CE3K's epic finale on Main Street. Although if anybody wanted to start something like that, we'd be down for it.
Instead, the film's fans tend to be film buffs who admire it for its storytelling and importance in film history. For example, the American Film Institute ranked the film 64th on its original list of the greatest American movies of all time. (Source) With that said, the same organization knocked Close Encounters off the list for its 10th anniversary re-release, replacing it with the likes of Toy Story, Spartacus, and Saving Private Ryan.
Other fans haven't been as fickle. Screenwriter Ray Morton wrote an entire book about its production history, and the film hasn't yet been bumped off the National Film Registry's list of "culturally, historically, or aesthetically significant films" to which it was added in 2007. (Source)
Even people who aren't fans of the film are fans of the film. That's because the film's most iconic moments have invaded our cultural consciousness like a virus, replicating and infecting other artifacts of pop culture. This had led many first-time viewers of the film to have a sudden sense of déjà vu.
The famous mashed potato scene was parodied in UHF, Weird Al Yankovic's love letter to all things entertaining. Of course, The Simpsons did it, too. In the episode "Homie the Clown," Homer sculpts a circus tent out of mashed potatoes. And South Park paid homage to the five-tonal phrase, where it was used to contact the Internet instead of extraterrestrials in the episode "Over Logging." If you've got an afternoon to spare and are tired of watching Survivor reruns, IMDb has a whole list of Close Encounter references.