Passports are not currently affected by the government shutdown. The passport service is, I think, not dependent on federal funding to operate.
I suspect that the rules here are the same as for USCIS services -- the majority of the budget for providing the passport service comes from the application fees rather than the federal government. Therefore a lapse in appropriations will not affect passport processing, save when they need to interact with agencies that are currently shutdown / significantly impeded.
According to the current passport office advisory notice:
We continue to offer passport services during the lapse of
appropriations for the federal government.
You can still apply for a U.S. passport book or passport card at all
passport agencies and centers and acceptance facilities (such as U.S.
post offices, libraries, or county clerk’s offices) during the lapse
of appropriations. You can also renew your passport by mail. Our
processing times remain the same: 4-6 weeks for routine service and
2-3 weeks for expedited service.
If you have a scheduled appointment at a U.S. Department of State
passport agency or center, please plan on keeping your appointment. If
you need to cancel your appointment, you may do so by visiting the
Online Passport Appointment System or by calling 1-877-487-2778. If
you have a scheduled appointment at a passport acceptance facility and
need to cancel your appointment, please contact the facility directly.
We will update this notice if there is a change in passport services
during the lapse in appropriations.