If you do not want to pay the cost of his unemployment benefits after you fire him, which is the only downside to firing someone in a non union situation (where there is no danger of an EEOC issue being involved, and where he has not just recently been on Worker's Comp, something like that,) then you should begin by giving a refresher to ALL employees about what company rules are regarding alcohol and drug use. Then, I'd suggest that you call in this particular employee, preferably on one of the occasions when you suspect he has been drinking, and warn him that "we have some reason to believe that you may have been drinking on the job. If this happens or is reported to us again, or we have any reason to believe that it has happened, we will be terminating you." He may throw a fit and jump up and leave at this point. So be it. No unemployment there.
He may wildly deny it, may demand to know who told on him, etc. You do not have to tell him anything. Just repeat that if you find that he has been drinking on the job, he will be terminated. Keep a record of the date you have this meeting with him. This is his warning.
Then wait. If he does mess up again, you fire him on the spot. Do not postpone the action, take it as soon as you believe he has violated the policy. It was his choice to change his behavior and keep the job, or his choice to take a chance, do it again and be fired. Thus he is not going to be approved for unemployment. He knew it was against the rules, he was warned, he did what he had been warned about, and he was terminated.
If he comes in obviously drunk and reeking and impaired, fire him on the spot, forget the warnings. This is considered gross misconduct, no warnings necessary, he is fired, and no unemployment benefits.
But if you just have something vague like somebody said they noticed he came to work a couple of weeks ago smelling like alcohol, you could go on and fire him now, without further ado, but the thing is, if he files a claim for unemployment, which he probably would do, and you have fired him without warning him and he denies that he did what you accused him of, and you let him keep working for another week or month after the supposed incident, he's likely going to be approved for unemployment benefits. This would cause your employer tax rate to go up.
The way to avoid this is to have firm policies about such things, give warnings and document those warnings, and then follow through with progressive discipline up to and including termination for violation of these policies. Get rid of your losers quick. They may mouthe off about getting a lawyer and suing your ass, etc. But in reality, they have no cause, they have not been terminated wrongly, and there's nothing they can do to you. But if you let them linger on the job, not only could they cause serious harm, not only could they fake an injury, or actually hurt themselves or someone else on the job, but they open you up to all sorts of liability as they represent your company.
I saw a case where this employee who was quite good at his job had smoked marijuana on the job for years. The employer was aware of it, and sort of turned a blind eye to it. Then he got angry at the man for something else entirely, and called him in one fine day and fired him for smoking pot on the job. This claimant actually got approved to draw unemployment. The employer had been aware of what he was doing, had never given him any warnings or indications that his behavior was unacceptable. So if you have an employee that "you believe" is drinking on the job, get on it and get rid of him/her.