Copper Mountain Ski Resort is about 100 miles from the Denver International Airport (DIA). The drive should take approximately an hour and 30 minutes. Flying into Denver is cheaper and a shorter drive than flying into Colorado Springs Airport (COS)!
If you do not want to rent a vehicle, Summit Express offers a shuttle from the Denver Airport (DIA) to Copper Mountain Ski Resort. They are only booking in the fall season right now so you will not be able to schedule this quite yet, but it is an option when the date gets closer.