They have a responsibility to abide by the letter of the law. Any other responsibilities you think they should have are your own personal ethics. Anything they do beyond the letter of law is what they may consider to be their responsibility and part of their ethics, which may have nothing to do with your ethics.
Businesses do not have to protect you from robbers and bad guys. The police don't have to protect you either. You can go elsewhere or you can secretly thumb your nose at the anti-gun business and if legal, carry inside, supporting the business while making your secret protest.
You're missing my point. I know full well that no one other than myself has a duty to protect me from violent criminals. My point is that private businesses are held legally responsible to provide reasonable accommodation for certain aspects of their patrons' safety such as fire protection, but not others such as violent crime and that represents a double standard. I fail to understand why a business should be able to be sued for damages if I'm injured or killed in a fire because they failed to provide adequate fire protection, but they're immune to liability should I be killed by a violent criminal when they not only failed to provide adequate security, but refused to allow me to provide for my own.
As I said before, I'm not in favor of private businesses being legally required to allow carry. All I'm saying is that if a business both bans carry on their premises and refuses to provide security of their own, they should be open to to civil liability just as they would for choosing to ignore any number of other potential safety issues. Conversely, if we're going to apply the "go somewhere else" standard to carry, then we should also be applying it to other safety concerns such as fire protection and clear walkways.
I also belive that with a very few very specific exceptions, "no guns" signs should carry no force of law for private businesses beyond a trespassing charge if the carrier refuses to leave when asked. The way I see it, if the law has no say in a private businesses' policy, then it also has no responsibility to to enforce those policies for them.