I started training in 1992, when I was 7. I hadn't seen Karate Kid, but things like Teenage Mutant Ninja Turtles, 3 Ninjas, those sorts of movies and TV (whose popularity can be owed to more seminal things like Karate Kid) certainly influenced it.
Now that I'm older, I really love Karate Kid 1 and 2. (3 and 4 were utter garbage).
As for kids in the dojo, at my TSD school, there are way more kids than adults (feels like a damned day care center). The Shotokan group I am a part of is based largely on college clubs and a few private dojos, and so that's entirely adults.
There's a dojo near where I work that I would consider "traditional" in its spirit, though they teach a mix of 2 Okinowan styles and JuJitsu. They have childrens' classes, but it's a small school and I don't know it well enough to say what % of the students are adults or children.
I'm in favor of children learning martial arts, that's how I started and I'm extremely grateful to have had it in my life from an early age.
So here's a question, how do child/adult ratios compare between styles? My personal experience is that TSD, TKD, and American "asian flavored" mcdojos have higher numbers of kids. Shotokan and Japanese/Okinowan are more adult oriented.