You have not mentioned whether they are dirty, or just smell musty..... If possible try airing them out first... I would hate for you to put them into a washer and have holes in the seams appear...
It has been suggested to put them in a top loader and let it fill up with water, agitate by hand, drain and spin and agitate by hand in the rinse water. Wet quilts weigh a lot and manipulating them too much strains the seams.... I would also use a soap made for quilts and remove it from the washer immediately and lay it on a sheet to dry or partially dry in a dryer first.
Storing them folded is not a great idea, roll them and use a lot of acid free tissue paper in the folds/creases. It is recommended to open them up and refold in a different manner every few months.