This is a mostly unresearched, uncited speculation on my part, so take it for what’s it’s worth.
Title II of the Civil Rights Act of 1964 states that "[a]ll persons shall be entitled to the full and equal enjoyment of goods, services, facilities, privileges, advantages, and accommodation of any place of public accommodation … without discrimination or segregation on the ground of race, color, or national origin.
Title III identifies twelve categories of entities that are defined as public accommodations provided their operations affect commerce. These entities include:
• places of lodging;
• places serving food or drink;
• places of exhibition or entertainment;
• places of public gathering;
• commercial sales or rental establishments;
• service establishments;
• stations used for specified public transportation;
• places of public display or collection;
• places of recreation;
• places of education;
• social service center establishments; and
• places of exercise or recreation.
Now, these sections do not apply to specific people, but rather to businesses that meet certain minimal guidelines, IIRC the place must employ 15 employees.
With these things in mind, I would say that it would be fine for a prostitute to refuse to have sex with a person who they don’t wish to. Your average working girl on the street is not bound by the anti-discrimination laws, and should have the choice in who her partner is.
The only time it may become an issue is in Nevada with brothels that have more than (again, don’t quote me on this) 15 people. If such a brothel is considered a “service establishment,” which I think it would be, it would be considered a violation of the Civil Rights Act for the brothel to refuse to service people on the basis of race (or for that matter, gender, religion, etc.). Thus, it would appear to me, that the brothel must have at least one worker who is willing to service anyone who might be discriminated against.
So, my off the cuff response would be that it is fine and dandy for prostitutes to discriminate on the basis of race, but a large enough brothel cannot.
*insert legal disclaimer here.