Answer: Usually it is sufficient to scrub the containers clean with soapy water and then dip them in a solution of one part ordinary household bleach/nine parts water, then let them air dry.
If you use a good quality soil-less mix formulated for container plants you should not need to worry about sterilizing the soil. The sterilizing (or sometimes pasteurizing) of soil is recommended under special circumstances such as when potting mix is recycled or you are using garden soil instead of a soil-less mix.
If you re-use your containers from year to year, I would certainly recommend using carefully cleaned containers and tools and fresh mix for seed starting and seedlings. Damping off usually occurs because the soil mix is kept too wet, the seedlings are too crowded, there is poor air circulation, and/or they are not getting enough light. So clean containers and soil can help reduce the chances of it happening but your overall care of the seedlings also makes the difference.
Many gardeners will re-use potting mix that is left in large outdoor containers over the winter, but they do refresh it by adding fresh potting mix or replacing the top six inches or so with fresh mix. Then they set their transplants into that. However, if you had a serious disease problem the previous year I would suggest starting over with cleaning everything (including tools) and using new potting mix as well.
I hope this answers your question.
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