I work for the fed. Be very careful not to mention the word alcoholic. Alcoholism is a disability and is part of the Americans with disabilities act (didn't check if you are in the US. If not, you learned something new). Base it strictly on performance. I would put her/ him on a performance improvement plan, if you have such a thing. This gives them written out what they need to work on and until when to show improvement. Hopefully, if she is having alcohol probs or the like she will get into recovery and cover herself from reprimands (since it is a disability). If not, and it turns out she is a bad employee, well then u hold her to the improvement plan and you have a reason to dismiss her.
Obviously your whole scenario maybe different than what I work with. But that is how I would handle it.
Good luck!
:-)