N.A.M. Rodger's Safeguard of the Sea: A Naval History of Britain 660-1649 has few short sections on docks in the 1400s and 1500s. The docks were constructed by digging an excavation into the shore of a area with large tidal fluctuations like Portsmouth or London. Initially these docks were closed with a dam of earth, which was economically feasible due the the cheapness of unskilled labor in that era. Gradually more and more timber was added to the docks until by the end of the 1500s timber gates that could be caulked watertight were added and the first true "Dry Dock" was created and the earlier docks were renamed "Mud Docks" to distinguish them.

The method that the most of the water was removed from the docks, (both mud and dry) was simply moving the ship in at high tide and letting the water escape at low tide. However the docks were also equipped with pumps, which had long existed aboard ships to evacuate the water in the bilges, and could be used to get the docks a bit drier than they would be by tidal action alone. I don't have any sources that explain how this process worked in less tidal areas like the Mediterranean. It's noteworthy that galley's and other more lightly built oared ships can be simply dragged out of the water and stored in galley houses when being repaired or simply not being used.