The simple answer is yes, of course they should, and it has to do with the very basis of freedom. To attempt to control the reasons for which people may or may not enter into agreements with each other, and by inevitable extension, to arrogate the power to determine, a posteriori, what those reasons were, is a serious attack on the freedom of the individual.
A few clarifications. Obviously if a booking has been accepted, a contract exists, and it’s too late to start adding conditions, unless the guest turns up in an unacceptable state. We’re not talking about that case. Nor is it a question, as some people have suggested, of allowing hoteliers to ignore the law. The question is about what the law should be. Also, if I were travelling with gay friends (yes, I have them, and I travel with them) and they were refused entry to a hotel or a bar for being gay I would be distinctly annoyed, but it’s a long way from feeling deeply my, and their, annoyance and discomfort, to making a law against whatever is deemed to have caused it. We are faced, every day, with people, situations, ideas, phrases, images, things, which we disagree with, which we don’t understand, which are disagreeable to us, which we would rather did not exist. To try to legislate them away is a possible last resort if it is absolutely necessary, but it’s a very bad place to start if we have any real regard for freedom.
Once the emotive idea of ‘banning gays’ is turned into ‘the freedom to decide who you do and do not wish to deal with,’ which is what the argument is actually about, it becomes a lot clear. Not as clear as it should be, though, because a surprising number of people seem to have the idea that businesses should only be allowed to function, on sufferance, subject to rigorous constraints, after they have abased themselves sufficiently in begging permission to be allowed to exist. People were freely exchanging goods and services, making all parties wealthier, thousands of years before the sort of interference we tend to think of as normal, and the governments that do it, were ever thought of. Such restrictions are unnatural and, beyond ensuring that contracts are honoured and that goods are more or less as described, they create severe costs to the seller and no benefit to the buyer, since all such restrictions raise prices.
All businesses restrict the nature of the goods and services they offer to those they have chosen to specialise in. Most then restrict themselves further for a great variety of reasons, usually, but not always, commercial in nature. Similarly, many businesses deliberately limit their customers, for reasons that are not always commercial. We normally see no reason for shops, manufacturers or whatever to justify the way they limit their own business. In fact, we rarely even notice. And just because our attention is drawn to a particular case, and we disagree with the motives given by the owners for running their business in a particular way; just because the very existence of that decision of theirs may be a source of inconvenience or offence to some people, we should not imagine that the right thing to do is to try to ban it.
Our freedom depends on other people’s. We should respect it, and understand the benefits it brings us.