Every Sunday pastors and others struggle with how to explain to unbelievers who are present why they should not partake. It isn't easy to be firm without appearing to be rude.
When I was in that position once, I came up with something I thought helpful, which I now offer to you. I've actually not heard it used by anyone else; you're welcome to it if you find it helpful, pastor friend.
We all know that unbelievers all tend to have a residual suspicion that religion is magic. How many times have you winced and cringed when a baptismal candidate says (s)he's undergoing baptism to help him be a better Christian—as if the ceremony itself, ex opere operato, zaps them with some spiritual blessing?
Well, that's how unbelievers regard religious ceremonies. They are magical, to them. They convey blessings. Crosses and Hosts repel vampires, don't you know? They have Power.
But we know that Communion itself has no magical power. It does convey a special blessing, insofar as we partake of it in obedient, worshipful faith (cf. John 13:17). But it does not convey any of this blessing to unbelievers. In fact, it can potentially convey severe judgment (1 Corinthians 11:27).
So why not communicate it that way?
Say something like this:
Helpful? It's yours.
Communion is like kissing a spouse: if it isn't your spouse, it really isn't for you. And so likewise, this is an institution that God intended for all Christians, but only for Christians. So we ask that only those who have been born again to saving and living faith in Jesus Christ partake of this service.
Now perhaps you feel that we are withholding from you something that might help you, make you happy, make you stronger. That is not the case. God is jealous of this ceremony, and the fact is that if you partake of this celebration of union with Christ, when you do not yourself have union with Christ, it will not help you. No, in fact it will harm you. So, for your own good, we'll thank you if you simply let it pass by.