If there is not a moldy smell I would use and try to get the smell out afterwards.
Batting inside bags may not smell.
Washing batting may come apart and ruin your washing machine.
I have been making quilts for about 50 years. I don't have a basement, I do keep a damp rid hanging in my fabric closet.
I mainly use high loft polyester batting and love it. I quilt on a 9" throat machine.