I'm not completely certain how old I was. My older cousin grew up watching TOS when it first aired, and he totally influenced me by having me watch it in syndication in the '70's when I was little. (He has a HUGE box filled with programs from the conventions he attended back then. I tried to snatch some but didn't make it very far...) My best memory is going to see STIV on opening night with him and his older brother.
Of course, when TNG debuted, I watched it from the first. I still remember throwing my pillow at the TV at the end of Best of Both Worlds, Part I.