When my kids heard that Leonard Nimoy had died, they said, "The guy from Mission Impossible?" They've been working through the classic '60s Mission Impossible series on NetFlix. It started at the same time as Star Trek, and as soon as Star Trek was cancelled, Nimoy went on to take the role of Paris in seasons Four and Five of Mission Impossible. They have yet to see any Star Trek.
As a kid, I watched old episodes of the classic Star Trek over and over again. We had about a third of them on VHS, and I watched those over and over again. When Star Trek: The Next Generation started, I watched most of that too. Some of my earlier memories of story critique: tearing down the writing of something, discussing how it didn't work, and how it could have been re-written as a good story spring from listening to my parents discuss how the (pretty universally bad) first two seasons of STNG could have been better, and by the time the bright point of seasons 3-5 was fading towards the dotage of seasons 6-7, I was old enough to join in these conversations myself. During the launch of Deep Space 9 I was probably at my deepest point into Trek fandom, but then when Babylon 5 launched my allegiance wavered and I eventually quit watching all the Trek shows in favor of B5. (By the time newer SciFi shows like Firefly and the Battlestar Galactica reboot came along, I had pretty much run out of time and stopped watching TV.)
So I certainly have many fond memories of Star Trek, but as of yet they haven't caused me to plunge the kids into those series. Some of that is perhaps that I've drifted away from science fiction in general over the last fifteen years, but another element is that I've pretty consistently found that the science fiction I was attached to in my youth has not aged nearly as well as the fantasy and other genres. As a young reader, I was deeply attached to the "juvie" SF novels by Heinlein and Del Rey. I tried to get our oldest to read them as she got into reading science fiction and fantasy and she shrugged them off, so I re-read them myself. I was, of course, not around reading novels in the SF "golden age" of the '50s and '60s, but there was a basic if slightly retro currency to Heinlein when I was 10-15. Rereading them a few years ago, I realized that age had set in during the intervening couple decades. Fantasy books written in the '20s through the '50s at times have a period feel, like a historical novel, if they're set in their own present day, but they are not that hard to follow. However, Heinlein's project future of the '80s and '90s as written in the '50s is far more alien and far harder to explain. Even re-reading the much more recent Ender's Game (a 1980s SF book which deeply affected me when I read it at the age of about 13) now reads as deeply dated: the warmed over Cold War tensions, the early internet projected into the indefinite future, the futuristic novelty of video games.
And this, I think, is also part of the problem addressing Star Trek from the distance of almost fifty years. (Next year will makr 50 years since classic Star Trek first started airing.) It's not just that the special effects now look incredibly old fashioned. It's that the vision of the future and the problems we project into our future have changed. Episodes that seemed to be important commentary at the time, like the plant with the never-ending war which has been turned into a simulated war in which virtual attacks are carried out, casualties are assigned, and those "killed" then report to vaporization chambers to meet their fate (all this so that the infrastructure of civilization will not be destroyed by conventional war), or like the endless fight between the half-black, half-white faced aliens, are now inexplicable and vaguely silly. In order to understand why they seemed important at the time, it's necessary to explain so much about '60s history and culture that you'd be better off watching a historical movie about the '60s and understanding it first hand.
In this sense, it's perhaps significant that one of the episodes that's held up best is a historical: City of the Edge of Forever, a story in which Kirk and Spock go back in time to depression era Earth. Because the setting is one that's now explicable, the story has held up much better.
Of course, there's also just the fact that City on the Edge of Forever was a well written episode, and many of the classic Star Trek episodes weren't. Out of the three seasons, maybe a third of the episodes were actually pretty good, mostly from the first two seasons. Part of what was so pioneering about Star Trek was that there hadn't been a for-adults semi-realistic science fiction show on television before. This was a new kind of storytelling and people were willing to forgive a lot because of that. Science Fiction is no longer the outcast genre that it once was, and as a result people don't have the tendency to put up with as many mis-steps.
There's something compelling about writing about an imagined future, and yet it seems like a genre of peculiar impermanence. I'm trying to think of a work of science fiction from more than fifty years ago that doesn't seem incredibly dated at this point, but although perhaps I'm missing something, nothing is really coming to mind.