Several folks stayed in Ogden for the Snowbasin phase of our big Utah Gathering last March. I stayed there some years back in the historic 25th street area. It's nice with lots of restaurants within walking distance. There are motels there with shuttle service to Snowbasin. Downtown Ogden to Snowbasin is about 20-25 miles, takes approx 40 minutes drive in fair weather. Ogden to SLC is about 50 minutes drive.
Some ideas if you are targeting Utah with MCP: you could go for one longer trip and ski Alta, Snowbird, and Snowbasin, or two shorter trips doing Alta/Bird separately from Snowbasin. In the first scenario a rental car would be very handy for relocations because the distance from Alta to Snowbasin is about two hours and logistically doable by public trans, but time consuming and with multiple transfers. The neat thing about MCP is that even after you use up your two free days, you get add'l days at half price, so you could add a day or two to a dedicated Snowbasin trip for fairly minor extra cost compared to total travel expense.
Sounds like you've committed to MCP and I'm not sure if that means driving to Utah, Aspen or Taos is no longer a consideration for you. There are reasonable motels near Aspen (Glenwood Springs) and Taos (town of Taos), but those would be best if you had a car, because they are 45-60 mins from the slopes. The decision to do a big cross country drive is a personal one. Some have more tolerance for that than others. Like some other posters in this thread, I've made that kind of drive to ski the Rockies numerous times. You do want to have a reasonably reliable vehicle, but I have done it many times with cars without snow tires or 4wd. The odds of hazardous driving on the interstate highways between Cinn and Denver is pretty slim for the few days all winter that you'd be traveling them. Once you arrive in ski country there can be a higher chance of seeing snow on the roads, but even then destinations like Loveland, Copper Mtn, Vail involve fairly low risk because they're only a mile from the interstate. In some ways, driving west from Cinn could be less challenging than driving to VT.
PS:
Oops, I should have paid attn to this from you before above post :-;
"Well, we ended up getting the Mountain Collective Pass. We’ll be flying to SLC and skiing Alta, Snowbird, and Snowbasin Jan 3-12. We can’t wait!"
We’re going to stay at a very affordable hostel in SLC from the 5th-12, when we will ski Alta & Snowbird. (Only $34/day.) We are going to look for an affordable place to stay near Snowbasin Jan 3-4. Any suggestions for cheap lodging near Snowbasin greatly appreciated!!