Hopefully if you do the job well and follow the suggested steps outlined earlier there will be no problems. However situations can arise.
The best way to handle customer complaints and problems, in most of the circumstances, is to remember that this is not just one job thing. If you decide to play the blame game things might get even more unpleasant. You have to decide then and there if being right is more important or having a happy customer.
There might be situations where you think that you chose the wrong kind of customer and you want nothing more to do with them. These clients are inherently problematic and you would do well to spot them before you start the job before you accept the job. But by and large you will be able to solve customer complaints man being accommodating.
For example, if a customer feels that changes need to be made even though they were not a part of the original design, are you going to argue about it or are you going to be accommodating? A little more of your time, geniality and patience and you will probably get a customer for life and hopefully any other reference through them. Sometimes that is more important than being right.
I have noticed a common flaw in many peoples dealing with customer complaints. They do not acknowledged the problem. Do not do that. Listen to what the customer has to say first. Satisfy the complaint with an explanation if that is what is required or make the changes.
However, some customers are problematic by nature. They have a tendency to either change the work demand in spite of of what was earlier agreed-upon or want more work to be done for the same amount of payment that was agreed upon. In most cases, declining politely and sticking to your decision in a stead fast but affable way best usually works the best.
If you still feel that what you are being asked to do is beyond your responsibility, decide whether you want to take care of it anyway. If you do not wish to do additional work for free, tell the client in a polite manner. If you do decide to rectify it just go ahead and do it in good spirit. Do not make the customer feel that you don't think it's your job but you are doing it as a favour. That is not going to create are beneficial long-term relationship with them. However, and do make it clear that this is a one time occurrence and you will not indulge the customer the second time.
That's that for customer care. Remember that your reputation is important to you. A happy client is often a source of many more jobs directly or indirectly. A happy customer will tell another of your great service but an unhappy one will will talk about his experience to ten others.