The United States Border Patrol is a federal law enforcement agency.
Pre-Apocalypse
Before the outbreak, the U.S. Border Patrol was responsible for detecting illegal immigration, terrorists and stopping terrorist weapons from entering the United States, and prevent illegal trafficking of people and contraband.
Post-Apocalypse
It is unknown to what degree, if any, the border patrol was involved in maintaining order during the outbreak. It can be assumed that officers reported to their stations to control the border crossing. It can be implied that any officer who stayed at their post was overrun, considering that the main border crossing in California was positioned between San Diego, California and Tijuana, Baja California- two heavily populated areas that were overrun.
At least three members of the Border Patrol were known to have survived and appear in Fear The Walking Dead: Passage - Colton and the two unnamed border patrol guard present in the tunnel under the border. All three were stationed at The Tunnel, which housed a bunker that they were making ready to occupy.