Alcoholism is a disease and is protected by the American with Disabilites Act which requires certain employers to afford reasonable accomodation for the illness. While I would not counsel dismissal based on being an alcohlic, you may be able to dismiss based on absenteeism, inability to work etc. I would certainly confer with a labor lawyer in WA to determine what rules apply to you in your situation. If you must dismiss the person, a salary continuance for a period of time would be a compassionate act.
Mr. Sarno is licensed to practice law in NJ and NY. His response here is not legal advice and does not create an attorney/ client relationship. The response is in the form of legal education and is intended to provide general information about the matter in question. Many times the questioner may leave out details which would make the reply unsuitable. Mr. Sarno strongly advises the questioner to confer with an attorney in their own state to acquire more information.
Alcoholism is a recognized disability. Firing someone for conduct resulting from a disability is illegal. Be very, very careful here. Before you do anything, I strongly suggest you hire a lawyer with experience in disability discrimination.
Disclaimer: Nothing I say here is legal advice. Do not do anything based on what I say. Hire an attorney to advise you. You are not my client unless and until we have both signed a fee agreement saying otherwise.
I did a quick check of caselaw before answering. I'm not so sure that alcoholism is protected by the ADA, which has set the bar as to what disabilities are protected that almost no one is protected. However, the Washington Law Against Discrimination has a much broader definition of what constitutes a disability. In response to a ruling by the Washington Supreme Court, the legislature codified an expansive definition. The problem is that caselaw says that an incident such as what happened with your employee is NOT protected, but all the caselaw is prior to the new definition of disability becoming part of the statute. You might be pioneering new territory. That's not a good place to be.
On the other hand, you cannot simply allow a person that you no longer trust to handle sensitive matters. You also have obligations to those who have entrusted you with sensitive information such as credit card numbers and you may be negligent if you don't take steps to protect them. The best solution for your problem will depend on a lot of things. Can you put this person in another position? Is there any other feasible reasonable accomodation? Also, how many people do you employ? (This will determine which laws apply to you.) There are many other details that may play into this situation.
I would really encourage you to meet for an hour or so with an employment law attorney to discuss the details of your particular case and plan an approach that will be compassionate, but that will also have the best chance of protecting your legitimate interests and avoiding a legal action down the road.
Good luck. I feel badly for both you and your employee.