Sometimes customers get upset. They may have been treated shabbily, or they want it one way, but it's the other. We have a culture of false apologies. There's an interesting article about the six types of apologies, with only one of them sincere. Most apologies are to get people to calm down and move along, especially in the corporate world.
It's difficult for a faceless corporation's communications director to show compassion when their company just did something atrocious. They're hardly in a position to take personal responsibility in any sincere way. Apologies are engineered and carefully worded, and vetted by the legal department. We don't have to do that in small business. Your most important asset is you, the person in charge, who built this thing and cares. Plus it's all your fault anyway, as we've established.
If you aren't sorry, the worst thing to do is apologize. If you cut a product line or canceled an event, well that's business. You can say "Sorry, but this isn't working for us as a business any more," but that's an explanation, not an apology, and it will be viewed as such. Disappointing customers is inevitable if you're around long enough and take enough chances. If you've never disappointed a customer, you're either lying to yourself or you have an extremely narrow line of business. Congratulations on playing it safe.
If you've legitimately screwed up, you should sincerely apologize. The article I mentioned calls this the apology from love. It's when you empathize with a person, putting yourself in their shoes, and put yourself out there. It's a real apology. You messed up, you apologize. What happens next is not important. You've done damage and it's up to the aggrieved to decide if it's in their interest to continue to engage you. I think the more you try to retain an aggrieved person in your apology, the more insincere you become.
Because sincere apologies are so rare, especially in business, people will often try to take advantage of you. They assume you're the corporate mouthpiece and now is the time to hit you up for free airline ticket or a discount coupon. Some will use your apology as an admission your business process is broken and their interpretation of how you run your shop should take precedence. I once apologized to someone in a store forum and they agreed to continue being my customer, if I gave control of that forum to someone else. A sincere apologize is an opening to some people, and you've put your heart on the line if you did it properly.
Here's where you need to be firm. "I'm sorry, but no," is what often happens when the opportunist sweeps in with their demands. In my mind "no" often means "get bent" or "GFY." My apology is not an invitation to become my business partner. And here is when you get to decide if you want to continue to engage with them or have them move along. The important thing is to stay firm to your convictions and your processes, provided they're good. You did your job, you've sincerely apologized, you've considered necessary changes. I'm always open to changing bad processes and a complaint will certainly trigger that change. However, many people just want it one way when it's the other.