Reputation among who? Everyone has multiple reputations at once - highly respected in some circles, highly despised in others.
Particular loud groups can make lots of noise about a business providing a platform to someone; but are those loud groups your customers? (If they are, maybe it is not your business anymore - they are going to tell you what to do simply by increasing the screaming volume.)
Looking at Facebook, IMHO, they get harmed more by arbitrarily banning people left and right, and then having to say “Sorry, we’ve made a mistake” and un-ban them. Not only it creates a bad optics, but it takes resources and human time, and undermines the trust into the platform.
You have to make those decision. I’m suggesting a base for such decisions - the law. It is a much stronger foundation than raw emotions and activist groups-influenced personal opinions.
It is also (IMHO) much easier to defend: “I can’t de-platform X because it would be discriminatory. I understand your concerns, and ready to support you providing that you form a legal defense fund in case of a legal action against my business if X decides to sue.” Vote with your money, justice warrior, not with your Twitter.