What constitutional provision governs the US border patrol?

Answer 1

Article 1, Section 8 of the US Constitution invests Congress with the following powers, among others:

To define and punish piracies and felonies committed on the high seas, and offenses against the law of nations; To declare war, grant letters of marque and reprisal, and make rules concerning captures on land and water; and is further empowered to oversee Immigration into the United States.

Pursuant to that power, the US Border Patrol was created by Congress on May 28, 1924 by an act of Congress passed in response to increasing illegal immigration.