A.S.
I grew up in Colorado and there is a small ski resort called Ski Cooper (not Copper Mountain) that is perfect for a family because it doesn't get as busy as the bigger ski resorts. Plus, the runs are smaller so it is perfect for learners. Yet, for the experienced skier it is still has fun and challenging runs. There is a bed and breakfast in town that we usually stay at called Tudor Rose. I bet if you contacted them, they would be able to set you up with a good babysitter.
I have found that Monarch Ski resort is pretty good for children and learners.