Safe deposit boxes certainly add a layer of security that is hard to beat, even with a safe in your home. For example, safe deposit boxes have a much better record of fire and theft safety than your typical home safe. However, they are not without costs and are certainly less convenient than a home safe. Additionally, safe deposit boxes have limited access to the person(s) listed on the account, which can make access after the incapacity or death of a holder(s) difficult and cumbersome.
If you determine that you need a safe deposit box, either due to the value or importance of items or documents, it is best to take the box as trustee of your trust, or name your attorney-in-fact listed on your durable power of attorney as an authorized person to access your box. If you take the box as trustee, any acting trustee can access your box, including your successor trustee(s) if your are incapacitated or deceased. If you name your attorney-in-fact that person can also access the box if needed, but you need to remember to change the name if your attorney-in-fact changes.
I have had several instances where the holder of a safe deposit box is either incapacitated or dead and their trustee or personal representatives cannot access the box without first going through legal and procedural processes with the institution where the box is located. Worse, usually considerable time, energy, and money is expended to gain access only to find that the contents of the box are documents that were valuable to the holder while they were alive, but had little, if any, value to the representatives or heirs. So if you are considering, or have, a safe deposit box think about what you really need to keep in it and who you will authorize to access the box.
PS, If you have a self storage unit, essentially the same advice applies.