You need to quietly ask yourself (and possibly ask the Lord) if these people are coming to you with a genuine concern, or just to bring you gossip.
Secondly, if your church teaches against homosexuality and has a "no-homosexual" standard for those who serve in ministry, and a number of people are bringing this to your attention, you do, indeed, have a problem on your hands. I assume that since this drummer is a servant in the ministry, he is aware of the standards of the church. Therefore, you should go to him, one on one, and ask him pointedly whether he lives this "alternative lifestyle." If I were you, I would not mention that other people have suggested that he is gay. That's messy, and it will probably make him very uncomfortable and resentful. Plus, he'll probably want to know "who" and you probably won't want to disclose that...
If he is gay, and comfortable, then you, your HR people, and/or your pastor need to determine what is the appropriate next step. If he is gay and uncomfortable, then you need to minister to him, and/or ask him if it's okay for you to talk to the pastor about this situation. If he says he's not gay, there's nothing you can do, IMO.
Regardless of what his reaction is, your only response to the people who originally came to you should be "thanks for bringing this to my attention; it has been dealt with," and nothing more.
Regarding the alcohol... big no-no. If you've smelled it yourself, tell him that you've smelled it, and ask him if he has a problem with alcohol. If he does, minister to him and offer outside assistance. If he says he doesn't, tell him that you cannot tolerate him serving in ministry carrying that odor, and that if it happens again, disciplinary action may be taken.
Most importantly, if I were you, I would tread very lightly. These are very serious issues that could really run him off if not handled with love and sensitivity. Watch your every word, gesture, tone of voice, and pray first.