You need someone to manage the motor.
Speed is extremely important for catching monsters so you need someone (or people) to manage the engine. They need to keep it oiled, do basic inspections and basic maintenance, and wiping, cleaning anything dirty. Anything complex they leave up to the engineer, but you need someone full time making sure the engines are at peak performance.
You need someone to manage the electrical equipment.
Ships have extremely complicated electronics, which are constantly being filled with salt filled air and giant monster attacks. You need someone to ensure that all the electrical equipment is functional. They'll help with ensuring all the advanced equipment you need to actually track down the monsters is functional.
You need someone to manage supplies.
You're gonna need lots of specialized supplies and rare materials. You need someone to source all the weird equipment you'll need, and supply everyone else with parts.
You need some expendable mooks.
You need a few people to do the minor manual labour that's not cool enough for the main characters. Second engineers, people to carry stuff, a bunch of stuff.
You need a cook.
People gotta eat, and so you need someone to make sure the food is nice enough for a long trip.
You need a medical officer
Injuries happen, especially when you hunt monsters, and you'll need someone to patch up the injuries and keep people alive.
Some of these roles you could merge, depending on how small the crew is, but you need someone doing all of them.