If you drive, then Steamboat might be a nice choice from MN for that time of year. It has a lot of moderate angle terrain that holds snow and usually skis well in early season. Most of the other major CO ski areas will also be reasonably good at that time too. Pretty enjoyable town too.
Is this pre and post New Years travel or more on one of the sides more? Too, by cold you mean Minnesota cold which is wet? Being from Iowa we've found the drier mountain air at same temp way more comfy .. almost balmy by some respects
I love Steamboat and if weather is like last, New Years should have nice coverage, whole mountain was open trail wise. However Steamboat Christmas to NY is can be pretty busy. I recall pondering this several times, went once and lodging was very limited if planning late. I recall once on mountain things didn't seem too bad and they are to have the new gondi up and running which should decrease any wait time at base significantly. Generally speaking I enjoyed it but we hit runs that were less congested.
Haven't done Whitefish so have no insight towards it. If doing Amtrak to Whitefish, then look for specials. Looks to be nearly a 24hr ride, leaves Minneapolis around 10pm and arrives around 9pm. I've taken Amtrak to Winterpark from Iowa, a shorter ride but will say when traveling with spouse and our middle, I opted to get them a sleeper on the return. Long story short, spouse hated coach but loved the sleeper. They get pretty pricey and often hard to find one, does include the meals though.
Cost wise Steamboat lifts are higher than many Utah destinations less you get advance w/lodging, typically one can get 30% off. If your 12 year olds ski free with a 5 day adult lift pass, but I'd double check on age limits at time of skiing. I know ours needed birth date verification of some type years back.
Not sure if you wish to fly but you have some direct flights to Boseman (Big Sky) and to Kalispell (Whitefish), price wise in comparison to Amtrak .. maybe not greatly different. I don't mind flying if direct. Course then, lots of other options with direct flights ...