We took our youngest at 14 months, and she's been every year since. My personal threshold would be once they are walking and able to interact with the characters, etc. I don't think I would want to take a small infant, but plenty of people do it. Taking a kid under 2 brings a whole different set of factors into play - you can't just go all out all day long, but have to work around the little one's schedule to some extent. Obviously every kid is different, but it's best to plan for daily down time, and you really have to plan and be prepared in the parks.
I know people like to say "wait until they will remember it", but I've never bought that logic. We don't just go to the parks for our kids to have fun. WE enjoy the parks, and in particular, we enjoy watching our kids have fun and making memories with them. No, she doesn't remember that first trip, but we sure do. Also - how can you not take advantage of FREE park admission at least once? We took ours 3 times for "free".
I was also surprised by just how much my little one remembered from her trips even at age 2 and 3. She's 5 now, and loves that place more than anywhere else on earth!