Dear Michael,
Yucca plants are typically found in the southwest in dry or well draining soil, full sun and not that much water. Without seeing the plant, my first thought is, it is rotting due to the overwhelming amount of rain we have had this spring.  There have not been enough days for the plant to dry out before the rain returns and if the drainage is at all slow, that may be the cause of your plant's problem.
I am assuming your plant is outside.  Is it possible/small enough to dig it up and pot it?  If so, use a sandy "cactus" soil you can find in the garden centers.  
In a few days the forecast is showing warmer days and the plant could dry out but again rain is also coming again at least that is the forecast today.  Covering the plant (depending on how big it is), would keep rain off of it directly but covers also cause moisture to stay put again rotting the leaves.  
Yucca roots can be huge and heavy and if it is not possible to move the plant, you could take cuttings and propagate them.
Yucca plants are pretty tough, you can take cuttings from the mother plant and grow them rather easily.  I am putting a link below for your review.  If you have further questions, please contact us again.