I was born in '65 and probably watched a few episodes in first run, but the show likely didn't sink in until I saw it in reruns. I do remember nightmares about "The Doomsday Machine." Most monsters one can run away from, at least for a little while. The DM could get you anywhere
Also, my dad was a high school English teacher and taught a special class using sci-fi literature. The class viewed a few TREK episodes, too, among which was "The Enemy Within." I remember playing with a reel-to-reel VTR from the school, crawling through the phaser shot frame-by-frame.