I started watching Trek when I was six years old.I think family show is a bit of stretch. It was dealing with some adult subjects. TOS was known for playing "what can we slip past the censors" especially when it came to women's costumes. DS9's war stories and things like Picard's torture aren't exactly stuff for the kiddies.

In fact everyone I know started watching Trek when they were either kids or teenagers. Compared to shows like South Park, it's quite tame.