Everyone makes mistakes. That humbling thought is one reason I work with music and sound and don't have a job involving life-or-death responsibility. I knew a guy whose job was regulating the oxygen that divers received; a mistake on his part could have had dire consequences I'd rather not ponder.
Even when lives are not in the balance, a mistake made working on a client's project can have unpleasant results. (A mistake made on your own project is a problem, to be sure, but at least it's just your problem.) Depending on the severity and obviousness of the mistake, the fallout can range from shaken confidence to months of work destroyed (“Erase hard disk? Y/N”).
So you know you'll make some mistakes, and that won't be good. Wonderful. How are you going to handle one when it happens?
A strong start is to do everything you can to avoid setting up situations in which mistakes can occur and to provide ways to recover from those that do. Plan in advance, confirm assumptions, document things assiduously, back up data religiously, save multiple versions of work in progress, double-check before committing to anything from which it would be difficult to backtrack, and most important, draw on knowledge of past problems to anticipate things that can go wrong.
Doing that much can actually protect you pretty well against many cataclysmic blunders — but sooner or later, something will go wrong anyway. That isn't fatalism, just a fact of life.
All right, so you made a mistake. Now what? Several responses are possible, and proceeding from there requires an important and sensitive judgment call as to which is appropriate.
Assess the severity of the damage: if you accidentally overwrote a console snapshot that contained a 0.5 dB EQ tweak on one track with a snapshot that's otherwise identical, it's possible the difference is so small that it's not even noticeable. (You may have made an unnecessary perfectionist tweak to start with.) If neither the client nor you notice the error, it might be best to just let it go.
Perhaps you can quickly and easily repair the mistake. In that instance, it is often possible to fix it without even letting the client know a problem occurred. I once witnessed an excellent engineer who was working on a major TV series accidentally fly an instrument to the wrong track, wiping the last bit he'd dubbed over. The client was engaged in conversation, and it was easy to re-create the earlier edit and then correctly execute the later one. It was done in minutes, and the client never knew — or needed to know.
But some mistakes matter and can't be covered up. That leaves only one option: give it to the client straight. As risky as it is making a decision not to mention a problem, lying about it is far dicier. If the lie doesn't work, the water can quickly get very deep.
On the other hand, if you concisely present the problem to the client in a calm, composed manner, what options (if any) exist for remedying it, and the cost in time and money of each possible solution, most clients will take it in stride and still feel the situation is under control in your hands. They won't be happy, but it may not precipitate a serious breach in the relationship or project.
Many times it is perfectly acceptable, after explaining the situation, to send the client away while you deal with things. That can help preserve the client's creative state of mind and momentum while sparing him or her from the concern that can arise watching a messy repair. If the mistake was serious, you might even have to end the session for the day, but even that can still be okay if the client comes in for the next session and everything has been fixed.
In the end, it is simply unavoidable that, to paraphrase a popular expression, excrement occurs. When it does, only two things really matter: your cleanup skills and your diplomatic chops.