Technically, the entire fourth amendment applies. BUT All the fourth amendment requires is probable cause for warrants, and that searches and seizures be reasonable. It doesn't require warrants for searches or seizures (although courts have found that that is usually necessary for reasonableness), and it doesn't require probable cause for searches or seizures without a warrant (though courts have found that that also is usually necessary for reasonableness.)
What the courts have allowed is the use of the border zone to justify exceptions to a lot of the things that are usually required for reasonableness. This isn't, technically, an exception to the Fourth Amendment, because searches still need to be "reasonable". Its just proximity to the border makes searches "reasonable" that wouldn't be anywhere else.
