Criminal courts are not shut down, but there can be some slowing in federal civil proceedings because furloughs if clerks etc which would slow down but not necessarily stop the law-fare.
Judges are considered “essential” employees. I strongly disagree. These people on balance have done more harm than good.
Furloughed employees typically get back pay and shutdowns haven’t saved money in past. Why they should be paid for not working is ridiculous. Never got paid when business shut down for snow days.
On the judicial side it is a little more complicated than that.
The Department of Justice has a major role in determining what is “essential” and what is not for federal courts.
If Bondi can find a touch of spine she can shut down most court operations.