I am not familiar with the commercial fishermen on Alaska's boats being unionized. They do have some union type rights such as injury protection, but for the most part, if you are dead weight on a boat, you are off.
Like any other job, getting on a commercial fishing vessel is either about being in the right place at the right time, who you know, or making a reputation for yourself. As you are asking this question, I am going to assume you do not know any captains to hire you and you are new to the fishing industry and do not have a work reputation yet.
The first thing I would recommend is reviewing this page. It is the State of Alaska's Department of Workforce Development page on the industry. Take some time and read all the pages listed:
If I were going to get a job on a boat, I would get up to Alaska to put yourself in the position of being in the right place at the right time. I would try to get work in a seafood processing plant. The work sucks and you'll probably be miserable, but it is the easiest way to get close to the fishing boat captains. Trident Seafoods is the major player in Alaska. Check their website for work and often they will spring for your plane ticket up there. I believe UniSea is another major player up there (sorry, have been out of the know for awhile). Some of the seafood processor jobs are unionized, but for the most part expect it to not be, expect repetitive work, and just basically it will probably suck.
The gold star will be if you can interface directly with the captains. Do your best to help them out without risking your job. Take notes for them, be friendly, do whatever it takes. Boat Captains are looking for people who are hard workers, do not complain, and it pleasant to be around as you are living in close quarters for long periods of time. If you run into them at the local pub after work, buy them a drink or buy their crew a round. You want to be memorable.
The state of Alaska does have regional job centers that are very helpful with finding work if things go bad for you. Sometimes the webpage I gave you lists boat jobs, but I do not see that too often.
So in short, you have to be in the right place at the right time. The difference between working on a boat and working the slime line at a processing plant is putting yourself in that position and making yourself memorable.