Can a person who lives out of state open an account in another state without ever being present?

What type of account are you referring to?

For instance, people open credit cards with the major issuers all the time via mail.

It would also depend on your bank's policy. Does your policy state that you can open accounts without the person present?
We specifically state that we will not do that.

Some banks will open loan and deposit accounts without the customer being there, and some will not. It is up to them. It is their choice. Internet banks and credit card issuers, as mentioned, do this routinely. It is in their business model. Many community banks wouldn't think of opening an account without meeting the customer face to face.