Judaism values life immensely and one must even trangress all Torah prohibitions (except murder, incest and idol worship) if that would save the life of one Jew, or even just extend it slightly.
As such, no effort is considered too much when it comes to finding a missing Jew who might still be alive but in danger.
Of course there are Halachic guidelines also as to how much to spend on finding the body of someone who is certainly dead in order to give him a proper Jewish burial. For example, one should not retrieve a dead body for burial at the risk of human life.
Hence the important question here is what happens when we don't actually know if the person is alive or dead - which of the above situations would apply then?
As long as there's still a chance that the person is alive, we must do all we can to try to save him/her.
But what about when the chances of the person still being alive are practically zero, eg. if the person was swept out to sea many days ago and there are no islands nearby and all beaches have been searched?
That is a difficult and painful question.
The tragic situation you describe is made even more tragic when the missing man has a wife. Jewish law forbids a married woman from having sexual relations with anyone but her husband, and if her husband goes missing and there's even a slight chance he might have survived and is still alive, the woman is considered an Agunah and is not allowed to ever remarry or have sexual relations with any man, ever.
Thus, the plight of this woman necessitates that we do absolutely everything possible in order to find conclusive evidence as to what happened to her husband. It is no longer just about being able to bury him properly (which is a grwat Mitzva in its own right) but also about the fact that proof of the husband's death would free his wife from a lifetime of imprisonment in a marriage which is no longer and would enable her to build an intimate relationship with another man.
We should leave no stone unturned in order to help this woman by finding her husband, even if it is almost certain that her husband is no longer alive.
I would like to use this opportunity to raise awareness to this very painful issue and to urge all Jewish couples to arrange a Halachically valid prenuptial agreement whereby if the husband is missing for a certain amount of time (eg. a year) and his whereabouts are not known, the marriage should become void.
This arrangement would save so much pain and heartache of so many Jews!
It's easy to think it would never happen to us, but the above situation of a man being swept out to sea, for example, can happen to anyone.
Please arrange for such an agreement, and please raise awareness of the very painful issue.
A similar arrangement can be made also by couples who are already married. Please speak to a reliable rabbi regarding how to proceed.