Interesting, the Wikipedia article has this to say
Mere customer satisfaction, or lack thereof, is not enough to justify a BFOQ defense, as noted in the cases Diaz v. Pan Am. World Airways, Inc. and Wilson v. Southwest Airlines Co. Therefore, customer preference for females does not make femininity a BFOQ for the occupation of flight attendant. However, there may be cases in which customer preference is a BFOQ – for example, femininity is reasonably necessary for Playboy Bunnies. Several breastaurants like Hooters have also used such requirements of femininity and female sex appeal under a BFOQ defense. Customer preference can "'be taken into account only when it is based on the company's inability to perform the primary function or service it offers,' that is, where sex or sex appeal is itself the dominant service provided."
So basically the question to ask it "Is it a bona fide occupational qualification that the driver be female?" Seems like a high standard to reach. Arguments based on "feels" as in "I don't feel safe around this kind of person/employee" seem like the very kind of discrimination that the law has tried hard to eliminate. It's pre-judging someone based on sex, and deciding that they aren't safe even though they haven't done anything. I understand that women are often harassed, but the law already has a process for dealing with harassment.
I predict this kind of thing (apps that allow customers to discriminate on the basis of protected class) will spread and eventually be challenged in court. Curious how this will all play out and become settled law.