^I love Shakespeare, and I love TOS. One of my favourite television programmes of all time. And I'm 26. I saw TOS at 5 years old, and fell in love.
You need an open mind, and a love of theatrical dialogue to get past a lot of TOS's production values and 'cheesy' effects (they were actually some of the highest budget effects available at the time, weren't they?), if you're under 20, that is.
I'm still struck at how ahead of the time the show was. The stories are miles ahead of a lot of TV shows these days.