It's easiest to wash the quilt in a top-loading machine. Start by filling the machine with water and add soap (I would start with Orvus). Stop the machine when it is full, so there is no agitation. Add the quilt. Let it soak for awhile, and hand agitate by pushing down on the quilt periodically. When finished with the wash cycle, advance the knob to "spin" and spin out the wash water. Allow the machine to fill with rinse water (at this point I would add vinegar to the rinse water), then again stop the machine to hand agitate, then advance the knob to spin out the rinse water.
Do not dry the quilt in the sun, as sunlight fades fabric. Also do not hang it on a clothesline. Water in a quilt adds a *lot* of weight, so hanging to dry puts a lot of additional stress on quilting stitches. You can dry the quilt in a home dryer if it fits. You can also lay a large sheet outside in the shade, spread the quilt on the sheet, then cover with another sheet or two (to prevent bird accidents) with ends secured with bricks or rocks. You can also lay the quilt to dry on carpeting; just place a fan on it to speed drying.
Sometimes soap and vinegar are not enough to get out all of a musty smell. If this is the case, what has worked for me with clothes (without ruining them) is to wash with soap and liquid Oxy-Clean 2X (got mine at Walmart). I wouldn't use dry Oxy-Clean, as I have found the particles can spot fabrics. However, the liquis Oxy-Clean seems to work really well in the washing machine to remove odors and it did not affect any of the clothing I washed that way.