What in the world has winter got to do with the OP's question. He asked, "How long would be a good stay at West Yellowstone to drive into the park each day and view the sights?"
I believe that the OP plans to stay in a campground in West Yellowstone, Montana. If that is the case I think you need to plan on at least 3 days and hopefully you are pulling a vehicle for sightseeing.
To me Yellowstone has two major loops. The Grand Loop - North and the Grand Loop - South. Driving from West Yellowstone, Montana and taking either loop will be about 125 miles round trip.
If you happen to decide to go to Grand Teton National Park I would suggest staying at Coulter Bay Village RV Park. Reservation are required. From Coulter Bay you can make a day trip to Teton Village which is a 104 mile round trip and you can also make another day trip of about 55 miles by driving from Coulter Bay by taking John D. Rockefeller Jr. Parkway south about 5 miles then getting onto Teton Park Road. Follow Teton Park Road until it re-joins Rockefeller Parkway and return to Coulter Bay.
Whatever you do, have a great trip.