If someone wants to go to Disney and haul a baby along, while I think they are completely nuts, it's their choice to make.
I think the real question is, "When is a child old enough to enjoy a uniquely Disney experience?" Let's face it...for babies who don't have enough information to process most of what they are seeing anyway, it wouldn't make much difference whether you walk them through the frozen food section at your local market or take them to the Market at AK. I don't buy the "oh, but he LOVES the colored lights and character" excuse. This is a parent trying to justify completely destroying the baby's routine.
Toddlers? I am not sure how developmentally appropriate Disney is for a toddler. They are old enough to enjoy...or be scared to death...by the costumed characters. They'll probably be entertained by parades and shows, but most of the rides aren't designed for such young children, and at this age they need to touch and taste to really understand. There are a lot of hazards for toddlers...they don't know enough to know what's dangerous or not, so all it takes is five seconds of inattention and your kiddo could be in for big trouble.
Personally, I think preschool is the earliest you can take a child to Disney and have them own the experience. Kids ages 3 or 4 and up can make sense of what they are seeing and hearing. You still have to tailor the trip around their interests if you want them to really have a great time, but it is do-able.
Which brings me to my next point: there is an age at which you aren't doing grandma any favors by hauling her to Disney with you. On our trip I saw a number of frail older people being hauled in and out of rides. You could tell they were weary beyond the point of enjoying the experience, and the fact that the temperatures in late May and early June topped out in the mid to high 90's, plus the humidity, didn't help.
I don't want flamed on this unless, like me, you have cared for an elderly, infirm relative in your home for at least 15 years.
So, my answer for all questions about when to go to Disney is what you'd find in the Bible...there is a time for every purpose under heaven. It is up to you to be wise enough to know whether this is that time or not.