Mount Rushmore National Memorial is open year-round with only the Sculptor's Studio closed from October - April. Visitors in the winter will find far less crowds with the Memorial seeing only 5% roughly of the visitors coming through the gates during December - March that would come through in July or August.
As for the weather, the Black Hills are a winter recreation destination but what visitors should know is that the region is quite temperate; especially compared to the surrounding plains in WY, ND and the rest of SD. Winter storms do occur but the highway from Keystone to Mount Rushmore is well maintained and provides easy access outside of more extreme weather events.
Overall, visitors to the Memorial during the winter season will find decent weather and almost all of the facilities and things to do available with no crowds.