I always just scrubbed my containers out with plain water between plants. If I wasn't planning to put another plant in a container right away, I would let it sit outside in the sunlight for several days. Spider mites don't live on the container, they live on the plant itself.
When you find an infested plant, a thorough spraying with soap solution every 5 to 7 days should take care of it. It's always done so for me. Spray the plant from every direction until the soap solution is dripping off the plant. Four or 5 treatments over the course of a month or 6 weeks should kill all the adult mites and any hatchlings from eggs the adults laid on the plant.
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Spider mites like hot, dry conditions, so raising the humidity around your plants can help prevent infestations. If your plants are small enough to handle, you can even hold them under running water from the faucet, rinsing them from the underside of the leaves as well as the topside. That will disturb any mites who might be thinking of setting up housekeeping on your plant.