Really you have five obvious options. I'll give you my two cents for what it's worth (and in no particular order) . . . . .
1. SLC - no doubt the skiing is great. You have three mountains to choose from, if you're into skiing a different mtn each day (ie driving to a different mountain each day). 45 minutes from the airport and can pretty much get in/out of SLC from anywhere and with little difficulty. Can be quite expensive - especially DV.
2. Steamboat - airport is less than 30 minutes. Obviously the airport is smaller than SLC but they do a VERY good job keeping that airport open and lots of direct flights. It's a different feel than SLC. It's one resort, everything is right there, very family friendly, not snooty at all. Awesome ski school for the kids. Great snow and skiing. With our four kids, we like to keep them in the same ski school or daycare everyday, so it's advantageous for us to go to the same resort as opposed to three different mountains each day.
3. Aspen - small airport with fewer direct flights and the airport doesn't seem as reliable as Steamboat (although I have no facts to back that up). Aspen is Aspen - the city is awesome but it has a reputation for a reason. If you're doing the family thing I'd probably recommend Snowmass which is right next door.
4. Vail / Beaver Creek - You've got the massiveness of Vail and the exclusiveness of Beaver Creek right next to each other. BC is "easy" and family freindly, but very expensive. Vail has everything . . . . and then some. Skiing is great at both. Airport is fine if you can fly direct.
5. Summit County - four resorts, a little further drive from the airport (90 minutes +/-). Flying into Denver, so that's reliable and will have direct flights from most places. Breck is the most popular - it's huge and has everything, however you're dealing with the altitude. If you want to do four mountains, in four different days, you may want to consider this.