link about banking regs for non-customers
While the check-writer's bank can verify that the check-writer's account has sufficient funds available to cash the check, there does remain the possibility that the person presenting the check (that would be you in this example) might be presenting either a forged check or a forged signature.
A forged signature should be obvious to the bank if they bother to check the signatory card of the account holder. Whether they do that is obviously THEIR responsibility, not yours.
A forged check (account number and routing number correct but amount forged) might be a little harder to detect, however. This would be the situation that I could see the bank wanting to not cash the check under. Think "Catch Me If You Can" type plot-line here. That is truly the only case where ethically I could see the banking institution legitimately being hesitant to cash a check written by a customer for a non-customer.
Having said that, from the Banking Institution's perspective, I can now see why the Banking Institution would be hesistant to cash checks for non-customers. This is where the situation now calls for DIPLOMACY, TACT, and SUBTLE PERSUASIVE TACTICS. You will lose your privacy with the teller in that you will become memorable, you may even leave a fingerprint behind, and your ID will be checked, but you will retain your privacy from the person who wrote you the check in the first place. This, I believe, is the whole point of cashing a check at the check-writer's bank in the first place, right?