Make sure the bank safe deposit box is rented by the Trust.
once i pass away, how can the successors prove they are the successors before they can retrieve the documents?
Typically the bank requests a copy of portions of the trust for their own records. Additionally in a well drafted trust there is typically a document known as the Certificate of Trust or Affidavit of Trust. It usually contains the basic Who's who of a trust. Copies of that document are typically accepted for ID purposes.