On a BSA risk questionnaire we use for new customers, one of the questions for the customer is what their intended use of a safe deposit box is. What exactly should we be asking here?

I dont believe I have ever thought of asking a customer this question. The assumption is that the customer will use the box to store valuables.
If you ask the question, I would assume that is the answer you will receive. I can't imagine that anyone contemplating to use it for illegal activity would readily tell you that on a questionnaire.

We haven't included this in our account opening due diligence. We aren't supposed to have knowledge of what is being put in a safe box, so I am not sure we should ask that question.

We do not allow non-customers to open safe boxes at my FI. They must have an account to do so...so hopefully we have a good understanding of our customer prior to them establishing a safe box anyway.

I agree with NewTooBSA- would the customer really tell you if they were hiding something prohibited in there?

That is what we thought. We were confused as to why it was on the questionnaire.

Fun little article:

http://www.redding.com/news/2009/nov/19/redding-woman-guard-linked-to-alleged-prison/

The girlfriend allegedly told investigators that the smuggling earned her $40,000 in profits in two years, and she had a safe deposit box at a bank filled with cash.

Drug agents served a second search warrant on the bank and opened the safe deposit box, which contained $8,020 in U.S. currency and 1 million Iraqi dinars.

"It was SINTF's first $1 million seizure," joked John Thulin, the task force's commander.

One million Iraqi dinars is less than $900 in U.S. currency

If the person tells you what they are using it for is one thing but I do not think you have any right to ask. Other than that it will not house firearms, drugs (illegal items) etc, which is stated in out agreement in legal ease. However, we did have someone tell us they were putting cash in to hid from Gov't and we filed a SAR