Why not:
* If you are short on boxes in any facility, then it makes sense to reserve them for your customers.
* The fact that a customer banks somewhere else, but wants to open a box with you borders on being a "red flag;" i.e. it simply doesn't make sense unless he's trying to hide from someone. (People usually hide from the IRS, creditors, or their spouse.)
* With no other connections, it's more likely that you will lose contact with this customer and he will simply abandon the box, causing you significant expense. (You could test this theory by reviewing boxes abandoned in recent years; i.e. did they have other relationships?)
* Safe deposit boxes are a "loss leader" for most banks; renting them to non customers doesn't change that equation.
* The most cost saving technique in safe deposit administration is to require the customer to have a checking account and require that the rent be debited to the account; i.e. there are no billing and collection expenses.
* Any BSA related "due diligence" information is based entirely on what the customer told you - you don't know anything else.
You are not required to treat customers and non customers alike.
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In this world you must be oh so smart or oh so pleasant. Well, for years I was smart. I recommend pleasant.