I'm going at the end of December and I'm hoping it is open by then. I would think that it will be closed during a shutdown. Even our local beach at Gateway National Recreation Area is closed though the city beach adjacent to it is open as always.
What I can't figure out is if there is no money to keep it open, where is the money to keep it closed and have someone posted there to do so.
There's a recent thread in the NorCal Regional subforum on this. Answer is "No, gate is closed and guarded at the park entrance." That point is at approx. 6,500' elevation, some 10 miles from the summit.