The most convenient way to get to parks with small children is the monorail. You can just push strolllers right on the monorail, no unloading or folding up like you do with the buses or cars. That being said, Contemporary is the least expensive monorail resort. However, it is a deluxe and you are saying you want moderate. What is the max. cost you can spend? If you go during value season, you can normally get a good rate at Contemporary wing rooms. Dining is spectacular right at the Contemporary, including Chef Mickey's character breakfast and dinner which is very fun for the little ones. You can also walk right to MK from Contemporary. It is about a 10-minute walk.
However, if Contemporary is more than you'd prefer to pay, I agree with the above poster regarding WL. During value season, the rates are very fair at WL. You can boat over to MK, which is a good transportation option with strollers and small children. The rate at WL will be lower than Contemporary, most likely. There are also nice eating options at WL.