Every port has equipped showers and washing machines. It's basically a whole infrastructure for getting cleaned up.

Access is by code, which gets radioed to the ship (or given in person after docking). If you know the code, everything's free. Apparently, you can sometimes find out the code at the cafe that's usually nearby.

But mooring the ship itself costs money. They charge our boat 130€ per day at the Amsterdam port. Not cheap.