I have worked for the postal service. My boss, who was an old timer, was issued a gun to protect registered mail at the airport until the times changed when people started going postal.
People rob and steal mail every day. That is why there is the United States Postal Inspector Service. The amount of information contained within a single mail truck is vast. People still get social security checks, pay their bills, and send all kinds of information through the mail. During the holidays, I can only imagine how many gift cards go through the mail.
Even today, registered mail is the most secure mail. Every bag has a special lock on it, which has a serial number as well as number that advances each time the lock is opened. All of the information about the lock alone and the contents of the bag is recorded every day at your local post office. If the numbers do not jog with the numbers on the lock, then a whole inspection takes place, and you can bet someone is being fired. Generally, a registered mail bag goes between a processing center and local post office. Since both locations keep records on the lock it is very easy to tell if it has been tempered with. The key for the registered mail bag is locked in a safe when not needed. The key alone is an accountable item, and the person checking it out, better not lose it as one key opens all of the locks. However, it is put into safe during the non-operating hours.
Now back to the exact subject of railroad post offices. The security of the mail was just as important if not more back then. In the 1960s and prior there were no electronic payments. Checks, cash, sensitive information about someone's social security number etc., all went through the mail. The priority of any postal person is to ensure the security of the mail. Even in a car accident, the mail must be secured before you check on the other driver. It depends on the situation, but if mail starts blowing into the street, the postal carrier must go get it.
Even today, terrorists target postal vehicles. Unbelievably, they wish to steal those vehicles, because no one questions those vehicles. Those vehicles have access to all government buildings and what not. If one of those vehicles got stolen just imagine how much explosives could go into the back of one, and how one could just drive into a federal building.
When I was working as a postmaster relief, I was told about how post offices are robbed, and what to do in that situation. You might ask how post offices could get so much money, and the simple answer is postal money orders. Even in a small town such as Capron IL, I used to do at least five postal money orders a day. As you might not know, you can only get a postal money order with cash or a debit card. Even certified checks are no good because a person can stop payment on them. Many small towns do not have everyday functioning banks, and I doubt they had them in the 1960s. So people as they did then, and still to this day, pay bills by postal money order.
I hope that answers your question. I honestly have no idea what UPS or FedEx does when someone steals packages from them.
On a side note, the fact that many do not understand how the postal service functions and the vital role it plays in society is disturbing.