It would depend on the state licensing commission. While prostitutes in counties have to register with the sheriff's department, legal prostitutes work in brothels, as opposed to Street walkers and bar flies. License fees for brothels run $100K to $200K annually. A potential prostitute looking for work in a brothel may be asked such a question and it is up to the owner to determine if he will hire her. He, or she, is ultimately responsible for any civil rights violations of the facility.
The majority of prostitutes in Nevada are unlicensed and work illegally in areas where prostitution is illegal: Vegas, Reno, Catson City. As they are already breaking the law, as are their customers, no civial rights action can be placed against them.
Lastly, most prostitutes aren't quite that picky. One girl told me a couple of months ago she prefers not doing black guys, but a Buck's a buck.
So until prostitution becomes legal in a majority of areas, and is opened up for independent workers, the question, while moot, would be no. It would fall into the same category as a waitress that refused to serve food to a customer in a restaurant. Repercussions to either situation if she boldly stated racial bias in refusing service could mean termination for just cause.
Caption this Meme 
