In "The Tempest", Act One, Scene One, why does Gonzalo believe the ship will not be wrecked?

What a specific question! Gonzalo first of all is seen arguing with the Boatswain who is definitely trying to cope with the storm.

GONZ:
Nay, good, be patient.
 
BOATS:
When the sea is. Hence! What cares these roarers for the name of the king? To cabin: silence! Trouble us not.

GONZ:
Good, yet remember whom thou has aboard. 

The Boatswain argues that the storm, the "roarers" of the wind and the sea, doesn't care who's on board - whether it's the king or not. Yet Gonzalo is still respectful of the king, and asks the Boatswain to remember his presence.

GONZ:
I have great comfort from this fellow. Methinks he hath no drowning mark upon him; his complexion is perfect gallows. Stand fast, good Fate, to his hanging. Make the rope of his destiny our cable, for our own doth little advantage. If he be not born to be hanged, our case is miserable.

Gonzalo then says the Boatswain looks like a man born to be hanged (i.e. hanged for committing some sort of crime!). His complexion (appearance) is "perfect..." for the "gallows". Therefore, Gonzalo hopes, Fate will not let him be "drowned", but save the ship in order that the Boatswain can fulfil his destiny - and be hanged.

It's the ugliness of the Boatswain that Gonzalo hopes will save the ship. Even if the ship was really weak, and leaking like a woman on her period (an "unstanched" - freely flowing - "wench"), then Gonzalo would warrant (bet against) the man drowning:

GONZ:
I'll warrant him for drowning, though the ship
were no stronger than a nutshell and as leaky as an unstanched wench.

Hope it helps!