A mattress dries slowly, so use less liquid than you think.
The Mattress Only Needs a Little Water
How to clean a mattress with a vacuum, a damp cloth, and baking soda, without soaking it.

A mattress can take days to dry all the way through. That's why the best way to clean one uses very little liquid.
Most of what builds up sits on the top layer. A vacuum, a damp cloth, and baking soda reach it without wetting the foam and padding underneath.
Strip the bed and vacuum the top

Take off the sheets and mattress pad, then vacuum the top, sides, and seams with the upholstery attachment.
Dust and skin flakes collect in the seams. Vacuuming them first keeps them from turning into paste when you add a little moisture.
Blot stains with cold water

Press a clean white cloth dampened with cold water onto the stain, working from the outside toward the middle. If water alone doesn't lift it, add one drop of mild dish soap to the cloth.
Rubbing pushes a stain deeper into the padding, and cold water keeps stains like blood from setting the way hot water can. Never pour or spray liquid onto a mattress, because it can stay wet inside for days.
Cover the top with baking soda

Sprinkle a thin, even layer of baking soda over the whole top and leave it for at least 30 minutes, or several hours if the mattress smells.
Baking soda draws moisture and odor out of the surface, and a longer wait gives it more time to work.
Vacuum the baking soda off

Vacuum the baking soda off with slow, overlapping passes.
Baking soda is a fine powder that settles into the fabric, so slow passes pick up more of it than fast ones.
Let it dry fully before you remake the bed

Leave the mattress bare until it feels completely dry, then put the pad and sheets back on.
Sheets over a damp mattress trap the moisture against it, and that is how mildew starts. A fan or an open window speeds up drying.
Short on energy?
Just strip the bed and vacuum the top. That's enough for today.
The larger lesson
A mattress is thick, and water goes in much faster than it comes out. Cleaning one well comes down to how little you wet it.

Blot, don't soak.
