How would you respond to a customer who wants to keep a gun in his/her safe deposit box? Any difference if it's an antique weapon?

Look to your Lease Agreement. Some have a list of prohibited items, others are silent.
Either way, technically you would have no way to know since you are unaware of the contents. Correct?

All of the various agreements I have used would permit the storage of the firearm. It would NOT permit the storage of any type of ammunition. But as previously stated, how would I know?

Review your lease. It may prohibit use of the box as a place to store firearms or inherently dangerous objects. In addition, you may have a sign on/at your front door or a state law that prohibits anyone other than a law enforcement official from carrying a firearm into your bank.

Whatever the prohibition might be, if the customer is discrete, you will never know it has been violated. However, in cases of severe testosterone poisoning, the customer may actually make a show of having the weapon, brandishing it in front of employees.

If you do have a published prohibition that is in compliance with the law of your state, you have the right to ask the person to leave the premises and secure the weapon before returning. If your safe deposit box lease contains a prohibition against storing firearms or other inherently dangerous objects, you may choose to deny access to the box.

The policy should be the policy without regard to the age of the weapon. A Buntline revolver is no less dangerous or frightening to your employees than a .45 automatic.

In addition to a prohibition on the storage of ammo in a safe deposit box, I have seen contracts allowing the storage of firearms, but only those that have been disabled (firing pin removed, for example).

The banker who showed me the contract limitation requiring that a weapon be disabled confessed that he would have no way personally to determine whether a gun had been crippled or how quickly it could be restored to working order.

ah, the things we put in contracts to give us comfort, regardless of the enforceability...