I was raised in a fundamentalist Christian family, and we were staunchly social conservative, so I can share that point of view.
So I got into Star Trek when I was 4 (thank you, mom) with the original series, and what had drawn me to it was the technology most of all: starships, warp speed, phasers, transporters, the whole shebang. I was obsessed with computers as a child (they weren't yet considered a common home appliance yet), and took every chance I got to see them on screen.
When the time Star Trek: The Next Generation came around, it was a shoo-in that I'd love the new whiz bang technology, but I was also old enough at that point to begin understanding the stories, and I found those exciting. I identified with Wesley Crusher at the time, even though he was some years older than I was, but it was great to see a kid as part of the crew.
Still, the primary focus was on the starships and the technology. I remember, in middle school, a friend letting me borrow his Star Trek Starfleet Technical Manual. I pored over those pages for days. I began to read Star Trek fiction, and write my own stories. Again, I was still socially conservative, but I was genuinely loving the Star Trek universe. There was never a time when I thought something was wrong or out of place.
I voted for George W. Bush in 2000. I went and saw Star Trek: Nemesis in 2002. It honestly just never came up that I was somehow not supposed to like Star Trek.