In populated cities, we carry dirty clothes ashore to a laundromat. We have an extra large army surplus canvas duffle bag that is perfect to carry big bundles of clothes. If there aren't any facilities nearby, we do our own washing. We wash them by hand with rain water using a set of buckets and a brush. The clothes dry well in the tropics, and during North American summers. Canadian winters don't allow for outside drying, unfortunately.
The sad fact is that all use of laundry soaps are toxic to sea life, even when bearing a label professing it to be "biodegradable" or "non-toxic". While the soap will eventually break down, it takes a while and can cause a lot of damage during that time. Phosphates, surfactants, triclosans, or any antibacterial ingredient, will do harm. According to the EPA, 28 ml (1 oz) of biodegradable soap needs to be diluted in 591 L (20,000 oz) of water to be safe for sea life. Washing clothes in plain water, with a good brush and a bit of white vinegar (diluted) can help freshen up clothes if there are no laundromats nearby, but in all, it might be better to wait to arrive at a facility to use soap.
For white clothes, or stained garments, we like to soak them in a bucket with some sodium percarbonate. It destains, deodorizes, whitens and is non-toxic to marine life. It breaks down to oxygen, water and sodium carbonate (soda ash) in your wash water.
When underway during long passages, we tend to wear the same clothes for a long, long time. If ever we do need to wash clothes, we'll do a quick salt water wash to save water, and rinse with fresh water. The clothes are hung out to dry on the lifelines, or on a separate line that we set up. Leaving dark clothes out to dry in the sun will cause it to discolour faster—such is the cost of sailing in warm places.