As someone who lives 271-391 miles (5 hours 32 minutes to 6 hours 32 minutes) to the Northeast depending on the exact route you take I can say that there has been an increase of avalanches in both Yellowstone and Grand Teton National Parks over the past 2-3 years.
There are not a lot of roads as Grand Teton National Park is more of a back country park and the main one Teton Park Road which runs from Moose Junction off US 191 going North out of Jackson Hole and Jackson Lake Junction with joins up with US 89/191/287 is closed from November 1 to April 30. Rule number 1 about road closures in the winter months around this part of the country is these dates can be extended. Personally I have seen enough snow to close a road in early October and roads not to open till the first to mid parts of June due to late snows.
My personal recommendation being from the general area would be to fly in 1-2 days early in order to get used to the elevation especially if you are from closer to sea level for elevation. The air is thinner and it will feel even more so in the winter due to the cold of the air. I would just relax or go to a museum in the area. Most of them tend to be small but they do have some of the local history and geology in them. At the end of the trip if you wanted to stay longer in the area I would suggest renting a vehicle and driving N on US 191 out of Jackson Hole to the Junction of US 26 where you will head East and will go through the Wind River Reservation eventually joining up with US 20 at Shoshoni (Show shon ee) where you will again head North. Stop for the Night at Thermopolis and enjoy the hot springs. The next day sometime between mid morning to mid afternoon continue N on US 20 to the Junction of US 14 where you will head West to Cody. There you will want to enjoy the Buffalo Bill Historical Center which is 5 different Musuems. In order to really see things you need at least a good day and a half. As you leave Cody take Alt US 14 towards Powell stopping at Heart Mountain Interpretive Center. This is one of the locations of the Japanese Internment Camps during World War 2.
For those with more time there are plenty of things to do in Both the Red Lodge and Billings areas including Skiing at Red Lodge Mountain, the Yellowstone Wildlife Sanctuary, Zoo Montana, Yellowstone Art Museum, Yellowstone County Museum, Western Heritage Center. Most of Montana has no sales tax though there is a bed tax for motels and hotels across the state as well as a resort sales tax in a few areas such as West Yellowstone and Red Lodge along with Big Sky and Whitefish have up to a 3% rate.
You can also take Alt US 14, US 14 or US 16 over to meet up with Interstate 90 and go into the Black Hills of South Dakota where there is plenty more to do outside of just going to Mount Rushmore. There is Chapel in the Hills, Custer State Park (some roads are closed in winter. Trust me driving the Needles Highway and Iron Mountain Road is enough of a challenge in the warmer months though it is fun to watch people driving huge motor homes through some of the tunnels especially the ones that have height restrictions. The fun game to play is will they fit and then to laugh at them when they take their ceiling vents off as they are raised enough that they don't fit. To date I have been lucky every time I have been behind one of these and not had to turn around due to them getting stuck though I am always prepared for that possibility), mammoth site at Mammoth Hot Springs, and more.