Duty
There's an old adage--or is it a warning?--save a man's life, and you become his protector; ever afterwards his life is your responsibility.
Apparently one corollary to this is: fish a man out of the ocean, and ever afterwards it's your obligation to save him from drowning.
Simply logic, really; he is the best swimmer of all of them, can dive the deepest and fastest. That's not pride or talent, just the natural legacy of growing up on ships, and all those years that he pushed himself, made sure no one would ever have to save him from the deadly waves again. And his legs are damn strong.
It isn't all the time. If Luffy just happens to fall off the ship--as he does, often enough--then it's whoever's closer, or whoever draws the short straw, depending on the direness of the situation.
But when they've all gone overboard, when it's an escape or an attack or a storm, when there's no time to debate or decide, it's his job, and they all know it, don't question it, take care of themselves and other concerns and trust him to get Luffy out.
He thinks sometimes that maybe he should mind it, but he doesn't. It's just one of the duties of a cook. At least on this pirate ship.