Mountain biking is allowed only on roads (no trails) in the park (with the exception of the Great Basin Visitor Center to the town of Baker Trail - 0.2mi). Helmets are strongly recommended. There are primitive roads to access for more of a challenge.
For more information on possible rides, check with a ranger at a visitor center.
The adjacent U.S Forest Service and BLM lands offer roads and trails for all skill levels of mountain bikers.