Back to the OP, I think they quite likely do influence behaviour in later life by enforcing gender stereotypes. It's been highlighted recently by reports of
Keira Knightley not letting her children watch certain Disney cartoons. I agree with her reasoning, although I haven't done any banning. I enjoyed reading to my children and enjoy reading to my grandchildren, I also enjoy sitting down with my grandchildren to watch a film. My grandson currently wants to watch Snow White when he's over, anybody that hasn't seen it in the last 30/40 years or so should make an effort to see just how cringeworthy it is now.
Snow White, helpless without a man ("Someday my prince will come") or dwarf for that matter (worth keeping in mind she was 14), falls in love with and is ultimately literally carried off by a man she had basically met once before and sung a song with, immediately takes charge of cooking and cleaning, abandons, with nary a backwards glance, the people that took her in and sheltered her and fed her in favour of looks and money. I'm also not keen on the dwarfs living in a shack in the woods, working hard down a mine all day digging up bucket loads of precious jewels while the prince spends his days living in a big castle or riding his horse around looking for girls to snog.
It's very easy to sneer at the PCness of it all and suggest that more is being read into things than are there, but it doesn't take any effort to watch them with a slightly different perspective and boil the stories down to their absolute basics, which is that a woman/girl is helpless without a man and her only hope is to find someone rich.