I'm not sure if you can do this with vertical blinds, but to clean my horizontal blinds, I do this:
I purchased a commode brush for use- lay them in the bathtub and fill with water and cleanser just to cover. A few swishes on both sides and they are clean. I then rinse and stand them up in the tub to drip dry. No mess.
You can put them in a pillow case and pop into a washing m/c on a warm wash and take out after the 1st rinse if you keep them in too long on wash & rinses they can become frayed at the edges.