He or she would only need to stay on board as long as there was a reasonable expectation that their expertise might help save lives
Once that moment passed he or she need not give up their lives for nothing
“Reasonable” in this context does not mean a more than 50% chance of success
but there must be a substantial likelihood (say 25% or so) that something positive will occur
There are many sources on this
One of the most famous is Ezekiel Landau’s (the Nodeh Beyehudah’s) response on  hunting in which he says that sport hunting is prohibited as too dangerous while if it is the persons profession he may hunt for a living

One should not and one is not obligated to commit suicide simply because they make a mistake. Perhaps in biblical times one could suffer the death penalty for a mistake against God. However, most judicial systems today consider each person innocent until proven guilty; requiring some type of evaluatory process before determining somone's fate. It would be foolish to think that the ship captain needed to go down with his ship simply because other people did as well. That statement strikes me more as vindictive and emotional instead of logical and legal.

 

In fact, the Jewish tradition places a high premium on perserving one's own life. A midrash recalls two men stranded in the desert. Both come across a jug of water. If they share the drink they will both also die since there is not enough to sustain them both. In this case the midrash advocates for only one of the men to enjoy the water since the preservation of one life is better than the loss of two together. Additionally, from a traditional perspective, suicide was deemed an offense to God, prohibited at all costs. Judaism advocates for each individual to sanctify life. And one can only sanctify life if they are alive to do so. In our modern era the Jewish stigma attached to suicide has subsided and allowed for much more compassion to those who would suffer such a fate.

 

This particular case doesn't even sound like the idea of martydom, which enjoys at least some level of debate. Many scholars suggest that the suicides that took place on Masada in order to dodge the capture by the Romans; and in the Middle Ages to avoid forced baptism are only somewhat justifiable at best. The context of those situations makes a great deal of difference.

 

That being said, rabbinic literature has occasionally dealt with the question of when it might be permissible to take one's own life. From time to time scholars have noted that if one were to force another to either: take the life of a 3rd person, partake in idolatry, or partake in forbidden sexual behaviors such as incest, then one would be permitted to take their own life and not commit those acts. The reason behind the permission here to allow for one's own death seems to be that it would prevent that same person from taking the life of another.

 

It should also be noted the high premium Judaism places on saving a life, pikauh nefesh. To do so, would be as the Talmud says, like saving an entire world. From a Jewish perspective it would seem that the captain of the Korean ship, as a result of being the captain, had an obligation to save lives before protecting his own. I would argue that losing his life in the process would have been a tragedy equal to the loss of any life. But I do not think that he would have been required to lose his own life in the process of saving others.