Until recently, I used the Swan and Dolphin business center when on property for work purposes (very expensive).
The best price I've found is at the Downtown Disney Hilton, which is a walk across the street from Downtown Disney. I last used the center a couple of months ago. Otherwise, if I'm staying on property, I use my laptop and go on dialup.
They used to have internet terminals in at least a couple of the resorts for .25 a minute. Not sure why they stopped doing that.