I think many 1970's and 1980's kids (outside of Japan) were exposed to anime but didn't know or realize it at the time. Beast King GoLion/ Voltron, Macross/ Robotech, etc. And then Sailor Moon and Dragonball came along and anime gained a lot more exposure. Followed by Pokemon and beyond...
For me, I think my first real experience was in the 1990's when I was old enough to purchase anime films like Akira and OVA's like Go Nagai's New Cutie Honey. Then, eventually the box sets of Neon Genesis Evangelion and Cowboy Bebop. :)