The official government website, found with a complex "passport services government shutdown" Google, says, as of now:

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.

Of course that may change.

The official government website, found with a Google search for "passport services government shutdown", says, as of now:

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.

Of course that may change.