We just ate there in Jan.. I remembered to dress nice-casual for our dinner as we were going from DHS to the Dolphin directly and not going back to our room to change. My Dh did not remember and I forgot to remind him. So he was wearing a Haunted Mansion t-shirt and knee-length shorts. They didn't say a word nor did they even sneer at what he was wearing. They were delightful and friendly. The place is so dark anyway (not in a bad way though), I can't see how others would really be able to see very well what someone else was wearing unless they are really looking.
We didn't have our kids w/ us this trip. We sat near the back of the restaurant and on our way through the place on the way out, we only saw one table w/ kids at it. I would bring mine with but only if I REALLY was hankering a steak because whew, I don't want to pay that much for my kids' meals which they would hardly touch anyway. For us, it was well worth the $200 meal (no alcohol).
Basically, unless you're worried about what others think of you, if you want to wear jeans, you shouldn't feel uncomfortable. Many people like to pass judgement on their fellow man but Shulas will welcome you.