One of the rules of the Darwin household (created with the primary intent of preventing master and mistress from becoming any less sane than they are already) is that the monkeys may not own videos or watch TV which the senior Darwins don't also enjoy. This definitely makes things like Barney, Seasame Street, Blues Clues, and anything with Barbie or Pretty Ponies or Care Bears or Strawberry Shortcake verboten.
However, every so often something sufficiently charming comes along that we genuinely enjoy adding it to the cannon. Such a one is Kiki's Delivery Service.
While we've enjoyed other Miyazaki films we've seen such as Spirited Away, they've been a bit too odd for the small ones to see without fears of nightmares and such. Kiki, however, is comprehensible to a 3-4 year old, beautifully animated, and just all around a nice story. (Maybe I should just break down an say "sweet", though without any of the treacle the term usually suggests.)
Kiki is a 13-year-old witch off to do the traditional one year's training alone in another city. With her black cat Gigi (a comic character who never goes over the edge into just being annoying like certain animal sidekicks one could name) she settles in a seaside town and starts a delivery service, flying around town on her broom. She's taken under the wing of a very pregnant baker and her husband, and makes a number of friends including a neighborhood boy obsessed with aeronautics, a young female artist who lives off in the woods, and a lonely old lady who likes to bake.
If you're looking for alternative to letting Poomba and Timon's Flatulent Adventure or some other product of modern American animation artistry into your home, give Kiki a shot.
Next up, how about a nice story about bunnies? How would a four-year-old like the animated version of Watership Down? Maybe not quite yet...