Here is the Wikipedia article about the legality of these Random checks.
A few excerpts:
...
The Fourth Amendment to the United States Constitution states that: “The right of the people to be secure in their persons, houses, papers, and effects, against unreasonable searches and seizures, shall not be violated, and no Warrants shall issue, but upon probable cause, supported by Oath or affirmation, and particularly describing the place to be searched, and the persons or things to be seized.” This fundamental right has a tense relationship with sobriety checkpoints. At a sobriety checkpoint, drivers are necessarily stopped without reasonable suspicion, and may be tested summarily and without probable cause. Thus the Constitution would prohibit people from being stopped without a search warrant or at least without probable cause that they have committed a crime; however, the warrant requirement only attaches should the search be unreasonable and the Supreme Court, as shown below, decided that such stops are not unreasonable under certain circumstances....
The Michigan Supreme Court had found sobriety roadblocks to be a violation of the Fourth Amendment. However, by a 6-3 decision in Michigan Dept. of State Police v. Sitz (1990), the United States Supreme Court found properly conducted sobriety checkpoints to be constitutional.
In the majority opinion, Chief Justice Rehnquist wrote, "In sum, the balance of the State's interest in preventing drunken driving, the extent to which this system can reasonably be said to advance that interest, and the degree of intrusion upon individual motorists who are briefly stopped, weighs in favor of the state program. We therefore hold that it is consistent with the Fourth Amendment."
I marked one sentence in bold.
The point is: The Supreme Court found that these checks are in fact constitutional.
Are we done here?