We spent two days at Disneyland with one-year-old Evie, and things couldn't have gone better. She napped in her stroller, didn't get scared, was great while standing in line, and thoroughly enjoyed the balmy 70-80° weather in Anaheim. While we had a lovely trip and I have no regrets about it, I don't think I'd take a one year old again.
I don't have enough cons to make a whole list, it's just that she didn't love it. While baby girl didn't shed a tear over anything at Disneyland, she really just didn't get it. And, the rides were very loud (something I'd never noticed before). She mostly watched, wide-eyed. I do think she enjoyed Small World. It was still decorated for Christmas, and she sang "falalalala" along with "Deck the Halls." We managed to get good seats for the Pixar parade, and she enjoyed that too.
Now, here are the reasons you should go.
If you enjoy going to Disneyland in general, then you'll probably still enjoy it with a one-year-old. We enjoyed going to Disneyland almost annually before Evie was born. The rides, the nostalgia, the food (churros!), the friendly people... it's just a magical place. And we enjoyed all of those things on this trip.
If your looking for a warm-weather getaway for your family, Disneyland is a great choice.
Anaheim in January was idyllic. It was deliciously warm, in the 70s and 80s. We were comfortable in tee shirts and jeans, just needing light jackets at night. Coming from freezing temperatures in the Salt Lake Valley, it was a glorious break from the cold. Since Evie is a December baby, she spent her first summer nearly immobile, or at least limited to crawling. By the time she started walking, it was freezing outside, and we were in winter hibernation. It was so fun to see her toddle around outside in her sun hat and tee shirt, and she loved it.
If you're looking for somewhere easy to vacation as a family, go to Disneyland.
Sure, you could vacation anywhere warm and take your baby along. There are dozens of public forum commenters out there that will say you'd be better off taking your baby to a shopping mall, but Disneyland is literally made for kids and families. Every bathroom has a diaper changing station, there are quiet secluded places to nurse, stroller parking is ample, and you won't find a dirty look even if your little one decides to pitch a fit. Disneyland is easy, and a much more friendly environment for a one-year-old than a museum or a swanky restaurant. It is also remarkably clean, much cleaner than a public park. The works thing our little toddler found on the ground was a leaf.