Correct. Through various means.

On the boats sailors do their own laundry, unless you’re an officer in which case your laundry is done by lower enlisted, and lower enlisted is comprised predominantly by minorities. The laundry folks also hustle on the side and will do your laundry if you pay them.

Pulling into port the Navy usually arranges some sort of pier side laundry service ran by whatever ethnic minority is being oppressed in that particular region.

Stateside there are laundry services on bases, and you pretty much nailed how they work.