What are the rules for driving at a public, uncontrolled airport in the USA?
The FAA has issued an FAA Guide to
Ground Vehicle
Operations -
A Comprehensive Guide
to Safe Driving on the
Airport Surface which covers what they call "non-towered airports"


  NON-TOWERED AIRPORTS
  When the control tower is closed or if there is no tower, the airport
  is referred to as non-towered. At a non-towered airport, you do not
  need controller permission before entering a runway or taxiway.
  Below are some best practices for operating on a non-towered
  airport:  
  
  
  When you approach the runways and taxiways, STOP, LOOK
  both ways, and LISTEN for aircraft that are landing or taking
  off. Vehicle windows should be open to do this properly.
  Alert others when you are using a taxiway or runway by
  always making an announcement on the radio before you
  enter. Be specific with your location and intentions.
  Always yield the right-of-way to taxiing aircraft and give them
  plenty of room. If an aircraft is headed toward you on the
  same taxiway, move out of the aircraft’s way.
  Always carry a radio tuned to the airport’s Common Traffic
  Advisory Frequency (CTAF) or UNICOM.
  If an aircraft is about to land on a runway that you need to
  cross, stop well clear of the runway. Continue to yield to the
  aircraft until it has landed and taxied off of the runway.
  Be aware that some
  aircraft at non-towered
  airports are not equipped
  with radios.
  Before you cross a runway, ensure that no potentially
  conflicting aircraft are taxiing, landing or taking off. Be
  aware of aircraft at non-towered airports that frequently make
  touch-and-go landings (immediately after landing, full power
  is applied and the aircraft takes off again).
  If your vehicle has a rotating beacon, be sure to turn it on
  anytime you are on the airport surface. Turn on headlights as
  well, being careful not to blind any pilots in the area.