I have spent a lot of time in Summit County, CO in the last few years. I understand not wanting to rent a car at DIA, but all the comments about being in Summit without a car are correct, especially if you want to get cheap lodging or ski more than one area.
The easiest way to go without a car is to stay in Breckenridge. There is a lot of lodging and a good local bus system. There is also a bus system that can take you around the county, but it may require several busses and a long walk, so if you want to go this way you need to be careful to select lodging near a bus stop. In general, it is set up to take people from silverthorne and Dillon to Keystone, and people in Frisco to Break and Copper. You can also go from Break to Keystone and A-Basin. Silverthorne/Hillon to Break or Copper requires a change in Frisco.
Another option is taking the A-line train from DIA to Denver Union Station, and then the Bustang from Union Station to the Frisco Transfer Center, or taking one of the shuttle companies to Summit. From there, you can get transportation to your lodging.
There are car rental companies in Summit, so you could come up and rent a car for the time you are in the mountains, and to drive to Dumont.
Colorado does have passenger car traction laws, and rentals may not have the required equipment. Here is the info:
https://www.codot.gov/travel/winter-driving/tractionlawAs for cheap lodging, there isn't much. The best prices I have found are the hotels at the I-70 Silverthorne and Frisco exits. Luxury Inn (pretty bad), La Quinta (OK), Days Inn, Super 8, Dillon Inn, Hampton at Silverthorne, Alpine Inn, Baymont Inn, Ramada, Holiday Inn at Frisco. I have sometimes found a good deal at someplace at Keystone. Also check VRBO, airbnb, etc.
If you don't mind driving, you could stay in Idaho Springs or Georgetown. If you really don't mind driving, try Golden.
John Unruh