They have the power to stop everyone in order to find out whether they have recently crossed the border. That's what a checkpoint is: everyone stops and answers a question or two.
What they do not have the power to do is conduct a warrantless search in the absence of any evidence that the person has actually crossed the border on that particular trip.
Sure they do - if they have probable cause otherwise. The last few times I went through these checkpoints (in Texas) they ran dogs around my truck looking for probable cause.
Whether or not you've crossed the border recently seems like fairly ineffective criteria. Are you sure that's the case? Where did you see that?
My experience: They had a drug dog sniffing at the vehicle without making it stop for the dog. They actually asked me to keep going, a little bit short of the stop sign, when I stopped for fear of running the dog over while it was very energetically sniffing in front of and at the sides of the vehicle.
The few times I went through them in the southwest (2009), I think they asked if I was a citizen and upon answering yes that was the end of it. They probably saw my uncooperative face, and didn't feel as entitled because it was earlier in the program. If my face had said "being hassled by uniformed thugs makes me feel safe" they would have probably given me a better molestation. And these days I would think they would feel more entitled since it's been going on for some time.
In reality any such stop is a detainment, and it is being done without any suspicion whatsoever. Exactly like drunk driving checkpoints (think of the children). But of course that's ignoring that our legal system has been turned to Swiss cheese by contradictions.
What they do not have the power to do is conduct a warrantless search in the absence of any evidence that the person has actually crossed the border on that particular trip.