Your question appears to ask about the vessels, the operator (company), and the captain.
Vessel. If it is an inspected vessel, you can verify its status by searching the Coast Guard's Port State Information Exchange database. This database also includes all issues raised during previous inspections and whether and how they were resolved. Note: You won't find uninspected passenger vessels (6-pax operators) in this database, just inspected vessels. Once aboard, you should be able to find a copy of the certificate of inspection posted.
Operator (company). There is no Coast Guard certification for companies that operate small passenger vessels. They inspect they vessels, and they license the officers, but they don't license or certify the companies.
Captain. Captains are required to have their original merchant mariner credential on board, but, contrary to a comment above, they are not required to post a copy. No passenger has ever asked to see mine, but I'd have no reluctance to show it if asked. I and the captains I know want our to customers to know we're legal. Note also that there's no guarantee that the captain you looked up in the merchant mariner database at home will be the one driving the boat when you go diving. Anyone with the appropriate credential can operate the vessel on any given day.
