Blacksmith
Blacksmiths must be highly skilled in metalwork and using a variety of tools in order to make repairs to the ship. They usually work with iron or steel, common materials in the construction of ships. Their skills may also be needed to make or repair weapons, or other small jobs on the ship. It is essential that ships employ at least one blacksmith, since it's never known if the ship will break down with no land in sight.
Boatswain
Without a good boatswain, a ship might not function at all. It is the boatswain's responsibility to oversee maintenance and deck activity over the entire ship, which is why junior officers are assigned to work under the boatswain to aid in these tasks. In relation to the crew, the boatswain must both keep up morale and make sure the crew works efficiently. In relation to the ship, the boatswain must inspect the ship daily, and make sure that all repairs are performed as well as possible.
Cabin Boy
Not always underage, a cabin boy is a member of the crew who has the task of running whatever errands the crew of the ship require of him. The captain's errands take first priority, of course, followed by those of other senior officers.
Captain
This is the big one. The captain has the most authority on the ship, and often is the ship's owner as well. The captain gives the ship direction and purpose, and holds a number of important responsibilities. The captain takes care of legal matters on board, and has the final say on the ship's destination, trade, and targets. A captain must also be a good leader, being the ultimate person the crew looks to in decision-making.
On a military vessel, the captain holds absolute authority over the crew. In a pirate vessel or otherwise more neutral ship, some crews may choose to vote the captain out of his position instead of the more violent methods of mutiny. In either case, the captain must make sure to keep the crew both happy and functional.
Cook
Any cook on a ship will have their work cut out for them. While the fare may not at all be equal to that of a restaurant, a ship's kitchen is just as busy, as the cook must prepare large quantities of food to keep the crew fed. To a lesser extent, they're also responsible for making sure the crew is fed a wide enough variety to foods to get the nutrients they need. A ship of any size will likely require more than a single cook.
Doctor/Healer
Whatever the title or methods used, this person is who the crew goes to when there are injuries or ailments on board the ship. On larger vessels, there may be multiple doctors with equal authority or a team of nurses working under a chief doctor. Anyone choosing this position must have the knowledge and skills to be versatile, and thus be able to handle the wide range of tasks assigned to him, everything from pulling teeth to performing delicate surgery. Most captains are loathe to leave this position unfilled.
First Mate
The first mate of a ship is the captain's right-hand man, and often takes control of the ship when the captain is unable to give orders. It's the first mate that takes care of the day-to-day problems, including paperwork, assisting in navigation, making sure the ship stays clean, and assigning the crew to tasks.
Gunner
There's no point in having weapons on your ship if no one can use them. Gunners are typically in charge of a group of powder monkeys, whose task it is to man the artillery when needed. A gunner's main duties are both aiming and making sure all of the ship's weapons and ammunition are clean and working properly. They must also train the junior officers under them in the use of said artillery. Most good gunners will require their officers to fire practice rounds often, even when there's no danger of engaging in battle.
Helmsman
Where the navigator plots the ship's course, and the quartermaster makes sure everything is running, it's the helmsman that actually steers the ship. They must be skilled enough in navigation to both read maps and use the right equipment to keep on course. It's a job that often requires long, boring waits while staring at the wheel, but a helmsman also has plenty to do in either battle or poor weather conditions.
Mechanic
For the more technical tasks, including taking care of fighter planes, a blacksmith or carpenter may not be able to do the job. That's where a mechanic comes in. They are at the boatswain's beck and call when it comes to fixing things, no matter if the job is large or small. While this position is most obviously required for military vessels, even a typical wooden airship would likely have a steam boiler and a radio, if not more that a mechanic would need to maintain.
As their title suggests, a navigator's duties are to direct the vessel. A navigator must know where the vessel's position is at all times, and be able to chart a timely course while avoiding any dangers that might lie on the way. A navigator also makes sure the navigation instruments are in working order. A good navigator is essential if you want a ship to go anywhere.
Passenger
Passengers are usually paid travelers on ships, though some do show a willingness to help out in the operation of the ship during their stay. These people often do not stay on one ship for very long, and, as long as their pocketbook is full, can usually find passage on any ship they choose.
Pilot
Unlike the helmsman, pilots on a vessel are assigned to both fly and man the weaponry of the smaller planes a ship may hold, whether their purpose be for battle or simply carrying passengers back and fourth. There are sometimes two pilots assigned to a single plane, depending on the model and size. A pilot is also expected to take care of basic maintenance for the plane, ensuring that it runs smoothly.
Powder Monkey
This is one of the jobs on a ship that can be done unskilled, but it's even more dangerous than that of the typical airman. Under the supervision of a gunner, it's the powder monkeys that work in groups to arm and fire the artillery. Depending on the ship, they may be treated better than their ancient counterparts, but that doesn't diminish the fact that they're often in the line of fire from enemy vessels.
Quartermaster
Quartermasters work directly under the captain and first mate, overseeing the operation of the whole ship. Quartermasters usually have a great variety of skills, most notably leadership skills, so that they may easily step in and take charge of a situation. They hold great authority, being required to gather supplies and dole out those supplies to the crew. They are also the disciplinary arm of the ship, maintaining order and giving out punishments for minor infractions.
Radio Ops
Nearly all modern airships are equipped with a radio to communicate both with other ships and any docks they might be planning to land at. Anyone taking this position would stand well to have a clear voice, and keep up to date on the ship's current position and if there's any certain rules or regulations for their current airspace. As you can imagine, they need to keep in close contact with both the navigator and the helmsman. For those ships that have a shipwide radio system to communicate with the crew, the radio ops would also generally be in charge of this.
Records Keeper
Paperwork. Lots and lots of paperwork. A records keeper is the ship's equivalent of a secretary, and must be highly organized to keep up with the practical matters of detailing much of what happens onboard. Other than the more mundane task of handling paperwork, a records keeper may also be asked to keep a more diary-like record of the ship's happenings.
Seaman
This is your generic worker you'll find on board every ship. On military vessels, they are often either of the lowest ranks or are simply hired workers with no rank at all. Regardless, these are the people who do the grunt work of manning the sails, cleaning the ship, aiding with repairs, being the bulk of raiding parties, and doing whatever other tasks the higher-ranked members of the crew require of them. Despite their mundane jobs, the sailors typically outnumber the ranked crew by a large margin.
Shipwright
A good shipwright is essential to keeping a ship running, especially ships whose hulls are mostly composed of wood. Even on metal-plated ships the shipwright plays an important role, whether doing repairs or construction, or even helping crew members with small projects. Shipwrights must be skilled in woodwork and the various tools associated with the craft. They are often given a small space on the ship in which to work and store tools and materials.
Strategist
It's inevitable that a ship run into dangers out there in the sky, and not all of them are as simple-minded as the flying beasts are. Anyone filling the position of strategist must be skilled in the art of tactics and logistics, especially in regards to aerial battle. There's the fair conduct of battle to consider as well, but most pirates you'll find won't care much for their strategist following that.