If you open your business to the public, you open it TO THE PUBLIC. Period. You may not discriminate against any customers. Legally, the only exception is if the (potential) customer is a threat to your business (e.g. drunken sot gets bounced out of a pub, angry ranting customer is making others uneasy, etc).
The right way to handle this is to make your business into a *private club.* Note the distinction from a *privately-owned* business.
If you are a club, you provide a special service to members. Think Costco. Augusta National Golf Club and the Boy Scouts of America are similar.
Anyway, the point is that if you have membership, you can restrict your services to members. Of course, you have to define how people become members and then stick to that policy. If you're not a club, you can't arbitrarily reject people, but if you are a club, the reject whoever you want. Problem solved!