In this scenario, the fishing vessel is a for-profit business operation and similarly to an owner needing to certify that a worker is legally able to work, a fisherman would need to ensure that his crew has the proper license to work. If a crew member was found to not have proper licensing, then the captain would be the responsible party.
Yes, a fisherman should be responsible for all of their crew members having proper licenses. Anytime a person has a crew of any type, they are responsible for the basic operation of their crew, including licenses, safety and legal procedures. A fisherman should be no different. Check with crew members to see if they have licenses.
The worldwide fish population is under strain, and over-fishing by many different companies is often considered to be the cause. To that end, I believe that it is important for all crew members on fishing boats to have the proper licenses so they are cleared to operate in a legal manner that won't strain the fish population further.
Although crew members should certainly be held responsible for proper licensure, at the end of the day, the buck stops at the captain of a vessel. So while a captain would likely require crew members to seeks necessarily licenses, she or he should also be responsible for making sure they have done so.