If you don't have motors, it's back to good ol' muscles, like Egyptians and Romans used to do. The only alternative is that if somehow there is another, intelligent enough marine species that could be trained, you could use those animals to pull the ship along the seas.
EDIT: TECHNOLOGY LEVEL
On Earth, the first functioning steam-propelled ships appeared in the 18th century, but they were prototypes prone to break down within a short time (often less than one hour).
Only in 19th century there would appear the first true steamboats. And they were affairs made more to cross rivers than oceans. Of course, given the pressure of obtaining a steamboat as fast as possible with no other options available, it's reasonable that the Watt engine is conceived even two centuries before than on Earth.
NOTE: Chimneys cannot be vertical tubes, or when you turn them on, first thing you cover the ship with soot. Ew. I think they should should be shaped like a car's exhaust tube, horizontal with the holes in the stern and additional filters to collect the soot and emptying it in the sea. (yes, very little ecological, but any civilization at its first invention will do ecological damage before improving their transportations)
Further considerations for this world
No winds at all implies that any kind of gas pocket will disperse by its own pressure along the surface, or rise in accordance to its weight, but limited by the pressure it finds.
In other words, a volcanic gas column will keep falling vertically on itself (after the main phase of the eruption has subsided) and keep spreading all around the volcano, permanently poisoning a growing area until much more land is lost for a very, very long time.
Any fume caused by the people activities (sewers, chemicals, even the animal wastes in the streets and in the farm) will stay where it is, progressively polluting the area to the point of suffocation.
All hot air will stay where it is, in pillars that would make flying for birds a near impossibility. In order to fly, an animal should be like a gigantic manta with low dense tissues and humongous floppy membranes to catch all the thermals they can in one sweep.
Plants that need impollinators would need land mammals to coat with their pollen. These mammals could look like tiny kangaroos so that they can hop faster in a field and spread the good stuff.
On a good note, viruses and bacteria that need currents to move would stay where they are along with their victims. Once their preys are consumed, they will revert to a sporal state so to be carried by passing animals or unsuspecting people.
Tall plants would also be very different: to transmit DNA and warn each other of biological attacks, they'd be connected to each other in a thick web of creepers, that would also serve to reinforce each other with nourishment. And people could use the creepers as roadways in this new upper-level ecosystem.
Swamps would be pots of pure undiluted poisonous CO2 and other nasty gases.
Deserts in daytime would be even more suffocating without any breeze to disperse the heat. presumably, they'd even be hotter.
Lakes would, well...stink! Not that their water wouldn't be drinkable, but you wouldn't want to stay in a place that stink of dead fishes and dead algae
Of course, you don't want to venture in a canyon without a gas mask, unless you like your deposits of CO2
Also, humanoids would much look like Voldemort! In a currentless words, no species needs sense of wind-carried smells anymore. They could breathe through gills that would also serve as air purifiers.
Every species' tastebuds would be extremely more refined to distinguish toxic from edible. Possibly, the standard tongue would look more like a snake's.
Predators should rely on augmented vision to better hunt. Instead of scent tracks, they's release a substance that leaves a visible mark at night.
I can think of only one reason why this world is not reduced in a hellish nightmare coated from pole to pole by a layer of heavy toxic gases, and that reason is: Bacteria. A form of microlife evolved to eat up all the nasty stuff and spew oxygen 24/7. On land, they'd appear as plants, on water as a thick plancton presence culled by fish so that it doesn't turn the oceans in a soup.