Chaosmancer
Legend
The number one thing that impacts the survival of the ship is how well the captain is able to steer the ship along the waves. You want to go into the storm, otherwise the wind and waves will slap into the side and flip the ship over. Especially if you have an empty hold. But, if you cut directly across high waves you cause the keel and hull to flex. This can crack the hull or split the ship. A skilled captain is needed to guide the ship so that it takes the least damage. In a really bad storm you'll probably lose a mast or two.
It would really suck if the captain was washed overboard...
Furthermore, in the larger ships there were specific sails that were used to guide the ship in storms. They were small, but still needed to be adjusted in the rigging. So there were sailors up there adjusting sails, tied off with safety lines to prevent them getting blown off. Most sailors who were on the deck tied themselves to the mast or the railing. A cubic foot of water weighs about 60 pounds, and several cubic yards of water will be crashing over the decks. Being able to minimize the overwash through skillful piloting makes it easier.
I agree with all of this, but this is one area where I actually will call upon the word of Verisimilitude (I feel dirty

See, if we hired a skilled captain and crew... then they have a plan to deal with the storm without our help. If the captain is washed overboard, then the First Mate takes over, and there is likely a person who takes command of the First Mate's responsibilities.
Additionally, adjusting the sails properly is a skill that many PCs won't have, and is something the crew is likely trained to do and knows how to do.
In many ways, I'm picturing this scenario like having a biker gang try and help out in a 4 star kitchen during the dinner rush. Unless they are shockingly skilled (have proficiency in water vehicles) all they are likely to be doing is getting in the way of the well-oiled machinery that is the kitchen staff, and causing more problems than they are solving.
Now, obviously there are ways for the DM to make this work, but the more the crew can handle the storm, the less invested the PCs are (because there is nothing for them to do). While on the other side, the more the PCs can do to save the ship and the people, the more strange it becomes that they hired people who seemingly can't sail through the area they said they could sail through.
Having someone that can go below decks and warp wood back into place, or a floating disk cast just outside a rupture could be critical. The cleric casting resist elements on the crew member steering the ship so they can handle getting washed with freezing cold water could be key. Flying up to the rigging would be an astoundingly bad idea, but having a fly on the thief as a backup in case they slip could be useful. The great strength of the fighter can help steady a line or a boom as the crew unfouls the necessary rigging.
Not a malicious point, but just noting... other than the fighter holding something, every example used magic.
So it comes down to
So then it becomes how can the PCs
- The ship is going to take damage, slowly, but perhaps catastrophically.
- If the ship goes down everyone's probably going to die.
- The captain (or who's ever steering) can mitigate, but not eliminate the damage.
- He might not be able to mitigate it enough, regardless.
- The crew can help the captain mitigate the damage.
- Some crew are going to be lost.
- If the PCs stay below decks they will be safe, unless the ship sinks.
- If the PCs help on deck, they are subject to the same hazards as the crew (which are bad).
- Help the captain steer the ship?
- Replace the captain to steer the ship if he gets washed overboard?
- Help the crew keep the ship responsive to the captain's orders?
- Maintain the endurance of the captain and crew?
- Prevent the captain and crew from getting washed overboard?
- Rescue the captain and crew if they get washed overboard (which will be hard to impossible in a storm)?
- Fix / repair the ship so it doesn't come apart beneath them?
These are good lists and good questions. My concern is just that, unless you are a sailor, I don't see how you can help with a lot of this except via magical means. Which means that the characters who can do things like cast "Protection from Energy: Cold" or "Enhance Ability" are going to get some mileage out of those spells, but it seems the biggest role of the party is supporting the people who were trained for this situation, which is an odd place to put them.
Not a bad place, and tailoring the situation to the players and thinking outside the box can lead to some cool moments, but it could also very easily slip into the party feeling like there is nothing they could or even should do, because they hired the best captain and crew, and those people know how to survive a storm.