Qualifications
There are several basic qualifications that you must meet before applying; being a U.S. citizen is just the tip of the ICE-berg. You need a valid driver's license and a two- or four-year degree, and you'll have to pass a bunch of tests, physical and otherwise (source).
Those without previous law enforcement experience must be under the age of forty. And even though it only relates to part of the border, you must be fluent in Spanish or display a willingness to learn—but you don't have to know French (take that, Quebec).
When you apply to the Border Patrol, they'll conduct a polygraph test, fitness test, background check, and a medical test. They'll also run a drug screening and check your dependence on substances (anything above "not at all" will get you dismissed from duty fairly quickly). Get ready for the six months it'll take you to pass all of their exams.
Once you've received a letter that you've been accepted, you'll go to the U.S. Border Patrol Academy in New Mexico. The training lasts for two months—four or more if you come in without being fluent in Spanish.
That extra time isn't a punishment, by the way; they just really want you to learn Spanish.
While at the Academy, you'll undergo physical training and a crash course in law. You'll also learn how to use firearms (very carefully). Even after training is done, you're still not completely an agent—you'll be stationed somewhere and begin your probationary period. The idea here is to prove that you can follow orders, display good judgment, and not accidentally shoot a fellow agent.
That's actually kind of important.