The Constitution does not impose an absolute bar on such profiling in immigration enforcement. In U.S. v. Brignoni-Ponce (1975), the Supreme Court held that Border Patrol agents erred when they stopped a car because "their only reason for doing so was that its three occupants appeared to be of Mexican descent." The pair did turn out to be illegal aliens, but the court held that the reason for the stop did not meet the threshold of "reasonable suspicion"–the same language the Arizona law uses. (Hat tip: SayAnythingBlog.com.)
Yet while the justices held that the men's Mexican appearance was not sufficient to justify the stop, they also said that it would have been legitimate to take it into account along with other factors.