If we are honest, we all have prejudices and biases. One of my biases is that I would object if one of my children wanted to marry a Muslim from the Mid-East and immigrate to that country. Whether my objections influence my children's choices in marital partners (should they ever actually decide to get married) is an entirely different question as is what actions I might take in such a hypothetical situation.
If you are in the U.S., then you should know that in fact, prejudiced people are allowed to conduct business transactions and do so every single day, with impunity. Engaging in business while prejudiced is not illegal. Excluding some individuals based upon certain criteria, including race, is illegal in most circumstances.
However there are limits. One limit is that if you are running a private club, you are allowed to admit or reject any members based upon whatever criteria you decide. So if you set up your brothel as a private gentleman's club, for example, you may exclude admissions to whomever you choose, for whatever reason you choose. However, if you operate a public establishment, you may not deny service based upon race, etc. etc. etc.
The question is: if you sell sex (which is not allowed in almost any part of the U.S. but frankly happens all the time, everywhere), are you allowed to discriminate based upon the race of the customer. Since prostitution lies outside of the legal business community, I am perplexed as to how any particular equal opportunity law could be applied and indeed, I wish someone would discuss why prostitution is being discussed rather than say: can a drug dealer decide to sell drugs only to members of some races? Or: Can a loan shark make loans to borrowers based upon the borrower's race and/or determine the terms of the loan based upon the borrower's race. I think the real deal with this thread is that some people get a particular kind of thrill figuring out ways in which women (because in prostitution threads, almost never are male prostitutes mentioned) may be forced to have sex with someone they do not wish to have sex with. In this thread, it's all under the guise of 'equal opportunity.' But IMO, it's the same old bullshit.
I am not in the US; and it is obvious that discrimination law, like all law, is impotent to regulate illegal activity, so where prostitution is not legal, pretty much anything goes (including prostitutes being forced to have sex against their will); the illegality of the transaction makes reporting rape more difficult, and makes securing a conviction that much harder too.
The question of applicability of discrimination law only arises in places (such as my country) where prostitution is lawful.
Obviously it is not illegal to be a racist; but as you point out, excluding customers from one's otherwise lawful business on the basis of prejudice
is illegal. Where that business requires a license (for example a bar), one of the penalties for illegal discrimination on the basis of race may be the loss of your licence - and hence your business. In short, as a racist, you may not own a bar.
The US rules that exclude 'private clubs' from anti-discrimination law are an oddity; IMO if you charge admission, and/or charge for the services provided, then there should be no exemption - but an historical distaste amongst legislators in the US for allowing Jews, Blacks, or other people they see as 'inferior' into their favourite bar has led to this anomalous and immoral distinction. That has no bearing on the morality of the situation; the existence of such exemptions is a moral problem in and of itself, and does not constitute an excuse for similar exemptions in other circumstances.
The question is: If you are going to allow people to sell sex, should you exempt those people to whom you grant a licence from the anti-discrimination laws that apply to every other business?
For drugs which are currently legal, such as alcohol or tobacco, refusing to sell them to blacks would be sufficient reason to revoke the vendor's licence. In no way is such revocation of license tantamount to forcing them to sell cigarettes to blacks. It merely presents them with the choice of concealing their prejudice, or not running that kind of business at all.
I am not aware of any anti-discrimination cases against prostitutes being brought before the courts here; but I am confident that if a prostitute could be shown to have rejected all of a large number of black men who approached her business, while rejecting very few white customers, a case could successfully be brought against her, and her licence revoked, and/or a fine imposed.