Here is what the USPTO says:
While activities across much of the federal government ceased at 12:01 a.m. January 20, 2018, due to a lack of appropriated funding, the USPTO remains operational.
This is possible because the agency has access to prior year fee collections, which enables the USPTO to continue normal operations for a few weeks. Should the USPTO exhaust these funds before a government shutdown comes to an end, the agency would have to shut down at that time, although a very small staff would continue to work to accept new applications and maintain IT infrastructure, among other functions.
Further information regarding any adjustments to the USPTO’s operating status during a federal government shutdown will be posted on the USPTO website and delivered to the news media.