The climax of Captains Courageous is when the boat Harvey is on wins the contest by filling its hold with fish before the other boats can do so.
Though Harvey has many experiences with the crew of the We're Here, the last and most important one takes place in early September. They've almost reached land so that Harvey can get in touch with his parents, let them know he's alive, and go home. First, though, they're going to participate in an annual contest to see which ship can fill its hold with cod and halibut the quickest.
This is the climax because Harvey is shown to be a different person at the end of his voyage than he was at the beginning. He's helpful, controlled, and kind. At the beginning, he was demanding, selfish, and petulant. Now he's the kind of person his parents can be proud of. When they win the contest, the change in Harvey is confirmed. All the work the sailors and Harvey himself did has resulted in a change in his character.