They don't have to check. They can even rent the car if the driver voluntarily tells 'em all this stuff. As long as the driver's not actually drunk at the time of the rental, you can rent to 'em liability-free. ("[Even] if the driver had informed defendant about his prior drunk driving convictions, the defendant rental car company would not have been negligent in renting to the driver in light of that knowledge.") As a matter of law. No jury trial as to whether that's reasonable. Summary judgment.
The contrary decision by a federal court in the Northern District of California in 2005? Wrong. An erroneous statement of California law.