I dunk and swish in a cupful of water with a drop of dishsoap, then rinse under fresh water, then lay flat to dry on a towel. I've been doing it that way for years and have never seen my straps get frayed or thin or compromised, though they may have lost some of their anti-flammability chemicals (which I don't care about, since I don't flick cigarette ashes on my kids, and their clothes aren't fireproof, so if my car goes up in flames they are in trouble just from that alone, I guess). The link in my siggy that goes to the technical encyclopedia ('here') says straps have been soaked in cola and juice and don't lose strength unless they are bleached and obviously damaged, so I've been using that as my guide for gentle washing throughout the years.
The manuals all say 'no submersion' in water, and only 'wipe with a damp cloth and gentle soap', or something similar, though. They don't want people to soak the straps in industrial cleaners and iron them (people are creative
, they will do stupid, stupid things) and then blame the carseat companies when they fail.