Why is Alonso feeling depressed and sad?

Alonso, the King of Naples, is feeling sad because he has lost his daughter Claribel to marriage, and now, shipwrecked on a deserted island, he believes his son, Ferdinand, is also dead. A loving father, Alonso says,

O, it is monstrous, monstrous: Methought the billows spoke and told me of it; The winds did sing it to me, and the thunder,That deep and dreadful organ-pipe, pronounced The name of Prosper[o]: it did bass my trespass. Therefore my son i' the ooze is bedded, and I'll seek him deeper than e'er plummet sounded And with him there lie mudded.

When Alonso says he thinks Ferdinand is "i' the ooze bedded," this means he thinks his son is dead. Alonso reveals how depressed he is when he announces he will "with him there lie mudded," meaning he wishes to kill himself.
Ferdinand, of course, is not dead. He has simply been spirited away, because Prospero wants him and Miranda to fall in love with each other.
Alonso, near the end of the play, repents of having helped steal Prospero's throne and restores Prospero to power. At that point, Prospero reveals that Ferdinand is, after all, still alive, which makes Alonso happy.