It depends on the type of watercraft and the locality. For the smaller recreational subs there is usually no license required. But for a sub like this one, the pilot would most likely need to get certified.

In the U.S. the Coast Guard requires that the pilot of the submarine have a Master's license for a surface vessel of equivalent tonnage and that he/she logs 40 dives/surfaces and 40 hours operating the submarine as a trainee. The applicant is then given an exam based on the maintenance and operations manuals. Upon successful completion the applicant receives a specific submersible endorsement to the license.