I was only four months old when TNG premiered; six years old when DS9 premiered; and eight years old when VOY premiered. I have a vague memory of once dressing up as a Ferengi for Halloween, though no idea what age I would have been at the time. I can remember waking up early to watch DS9 and VGR; the local syndication airings were on Sunday mornings for a while.
In any event, I watched the TOS movies more than I watched any other Trek while I was young, even including DS9 and VGR (I wasn't very good at waking up on time for the shows, and I'm pretty sure the local channel airing them bounced the showtimes around). Later, the TOS movies led me back to watching TOS in syndication, and eventually that led me to watching TNG. I never saw most of DS9 until a couple years ago when I bought the DVDs, though I had seen most of TOS and TNG by that point.
Short answer: I think so, since it was mainly the TOS movies that led me backwards to TOS, rather than the various spin-off series. However, since I was watching those movies and
some TNG, DS9, and VGR, it's difficult for me to separate it all with certainty.