Nobody likes to screw up. But your employees should not be
afraid to bring problems to you. If they do not feel free to bring problems and discuss them, it will take longer to discover and fix problems, and the problems will
fester in the meantime.
The key here is to look at how you react to problems. If your first impulse is to attack and blame, you really need to work on that. Make sure that your reaction to problems is positive and resolution-oriented.
Ask questions and make sure you get the information. Your first response to needs to be to contain the damage, and that is not going to happen if you reduce your employee to a quivering puddle of fear. Find out how bad it is, and what your employee thinks happened.
From there, communicate out to affected people. I usually ask someone who is on the team but who is not an expert in the technology in question to take the lead on communications and notifications. You should notify your own boss. That leaves the technical experts free to concentrate on identifying and troubleshooting the problem.
Set up a bridge line for communications. Don't let it be captured by people who are asking unanswerable questions. When that starts to happen, break in with a situation update and the current state of the resolution plan. (Sometimes, that may just be a list of the possible causes that you have ruled out.) Make sure that there is a sense of forward motion and progress. If people insist on asking unanswerable questions, tell them that there will be a postmortem to discuss the causes of the problem and the long-term resolution plan.
Part of a manager's job is to make sure to fly air cover for your technical staff when they are working a problem. They can't concentrate on a resolution if they are constantly being asked "are we there yet?" Ask them for regular updates, but all other communication and updates should flow through you. If necessary, set up a separate bridge line for technical experts and keep the non-technical people off of it.
After the immediate crisis is over, hold a postmortem session to dissect the problem, the team's reaction to it, and how to avoid the problem in the future.
Then, if it is necessary to reprimand your employee, do it humanely and professionally. Reprimands should be direct, clear, private, and over. Don't let the problem drag on, and don't let it fester.