What is a business owner to do when one of his employees has been alleged of a crime and the media has publicized the business and employee name?

Enter your answer issue a press release and try to solve the problem, you can use press release services or ask a reporter of your local newspaper to do a story on the solution.
Bad press and bad things can happen, it is the manager job to do crisis management, repair the damage, and regain reputation

I know this situation well. It can be awful. You have to take strong and immediate action to show that you or your company haven't been complicit in the alleged crime or that your company didn't attempt to hide or cover up anything. You said the allegations are likely false. Without knowing the facts it's difficult to say, but you have the tough decision of whether you distance yourself from the employee, which is likely the right thing to do for your company. You want to stay out of the media and hope it passes soon. This means being silent at appropriate times, but speaking up when necessary to clear your name. I'd be happy to hop on a call and discuss if you're still dealing with this problem.

Although not specifically a PR person I've been involved in mathematical model of media (including PR). I've also been involving in hundreds of hiring decisions in the financial and other industries. (The financial industry has significant licensing requirements that are specifically concerned with this situation. They also train all employees on media relations. I have also worked in enterprise risk management for a major bank.

You are facing three different business risk issues: PR (aka reputational risk), legal liability (lawsuits from non-employees) and HR risk (lawsuits from your employee, loss of employee productivity as he/she deals with this issue, as well as potentially licensing issues depending on your industry). You need to consider these in parallel.

Let's start with the HR and legal issue before even moving to the PR issues. You need to discuss the HR side with a good HR person. You will also need an attorney in the loop for a number of reasons, including advice and conversation confidentiality. Obviously, you'll want to given your employee benefit of the doubt and presumed innocence. However, he or she may now experience sub-par job performance due to these personal issues. Also by retaining this person you may be accused of negligence or vicariousness, particularly if the person holds a position of responsibility. Obviously, there may be PR fallout which we'll consider in a moment. In some industries, there are licensing requirements, and state or federal regulators may help make some decisions for you. These are issues your HR and employment law attorney will need to sort through. Note that a number of firms provide HR outsourcing services, so you may be able to retain top-notch HR expertise even if you lack it in house.

The PR side is much simpler. As it turns out, with unlimited financial unresources, academic models suggest you can pretty much spin anything anyway you want. (The bad news: of course, most businesses don't have unlimited resources. It becomes harder to must financial resources if the firm is also fighting off lawsuits from employees, or customers, losing sales in the immediate aftermath of the incident, or is itself fending off investigations by regulators or law enforcement. Under these kinds of conditions, reputational damage can sink a firm. This is why companies invest so much on trying to hire the right employees as well as carefully manage compliance and reputational risks. The idea is to try to prevent these kinds of crises from happen in the first place.)

There are generally only two distinct states to consider: (1) effectiveness of PR for given $$$ with the person remaining employed and (2) effectiveness of the same PR spend without that employee remaining associated with the company. (There may be circumstances where there are other states. Some government agencies may have to first suspend the employee.)

This is where a huddle with your HR and legal risk advisors will come in handy. They can estimate the risks of legal or regulator action from your employee or customers in both of those two conditions. You can then weigh that risk against the effectiveness of PR in the two different conditions.

In general, PR is in many ways similar and complementary to advertising. (And running an advertising campaign in the immediate aftermath of this incident is one possible response.) You can estimate the impact of PR from what you already know about how consumers respond to your advertising messages. Although what I said earlier is certainly true (for an unlimited amount of money you can spin anything to your advantage), for limited amounts of money you may note that advertising (and PR) are not always as effective as you'd like them to be. You need to balance your expected legal risks against your expected PR & marketing in the two different condition I've described.

One other point: I mentioned that the financial industry trains all employees in media relations. After discussing with your legal & HR advisors, you may want to put in a crash training program for your employees (aka company email.) On hire, many financial employees are told to refer all media contacts to company media relations. Financial employees are often told at hire that unauthorized discussions with the media are potentially grounds for termination. Of course, different industries have different conventions. It's also important to keep your employees happy during a crisis. (If employee morale is a concern, there are crisis management firms that specialize in crisis employee relations. I have seen "LiveStrong" type bracelets passed out and employee loyalty oaths signed, with unclear effectiveness.)

Legal disclaimer: I'm not an attorney. You will definitely want to seek professional legal, PR, and HR advice in this situation. If the suspected crime involves on-the-job fraud, you may also want to consult with a forensic accountant.

I'd be happy to discuss my knowledge of marketing and managerial decisions in greater detail over a phone call here on Clarity. I'd also be more than willing to give advice on screening employees and putting in place adequate enterprise risk controls to try to avoid these issues in the first place.