It would depend on WHY it is getting muddy. Is the water draining in from outside? If it is, a French Drain will help if placed correctly. If the water is coming up from below due to high water table, it will not.
Yes you can. It is used for in places that accumulate excess water and will help in keeping it dry. The French drains will also stop and reduce any mold or mildew growth. In addition to those benefits it the drains will also help prevent structure damage caused by water.
French drains can be a solution to this problem. However, you must realize that it is not a guaranteed fix and you still may experience a muddy crawl space after rain or heavy snow.
Well French drains are designed to take water away from building foundations so in heavy rain and snow I would think that this action could solve your problem and also stop any possible rotting happening.
Removed whatever dirt necessary to crate a positive slope way from the house. Leave a minimum of 6 inches of foundation exposed. The ground should slope 6 inches away from the foundation. The hidden piping and drain system can trap water before its enters the house.