Much of the park is at a high elevation and snow can occur at any time of the year. Check with the park website or at the visitors center for any road closures within the park. Part of the Scenic Drive was closed when I visited.
Great Basin National Park is off the beaten path and I do not think there is any mass transit coming here. You need to use a private or rental vehicle to get here. The nearest airport is about 68 miles west near the town of Ely.