I'll fly to Billings or Bozeman, probably by United, rent a car, and stay at Mammoth or in Gardiner. The road between the North Entrance and the Northeast Entrance is open all year so people in Cooke City and Silver Gate can get out (Beartooth and Dead Indian Highways close for the winter).
Salt Lake City would be a last resort, and probably cheaper, but that makes it hard for a three-day weekend, or four if you consider I'll probably fly out on a Friday.
"So...now you see that evil will always triumph, because good is dumb."— Dark Helmet