I cleaned carpet for about 6 years with my uncle who owns a business, so I will give you the easy way to clean.
I am assuming you dont have any bad stains or things like that, which require a little different type of work.
1. Take it to the car wash and use the soap setting on light pressure, sometimes using full power on some dirtier areas. After washing with the soap setting…switch to rinse and go back over everything. Do the best you can to get out all the dirt as well as the soap. You dont want to leave the soap behind, as it will attract dirt again to that spot.
2. Vacuum the carpet with either the one at the car wash…great suction by the way. Or use a shop vac at home. I would use an ice scraper to pull out any excess water into the drains in the boat..again..vacuum.
3. Let it dry completely. Dont get silly and play around in the boat while the carpet is clean, but damp. You can set it in the sun, or put fans on it to speed up the dry time. If you really want, buy some cans of Scotch Gard and spray the carpet down. This will help repel some staining for a little while. There is a lot of traffic in a boat, so that will wear faster than usual.
Again…make sure anything you do to the carpet dries first, before you go playing in it.