Let 'em dry. Just leave them in your tank's cabinet so they don't get too dirty. A little bit of dust won't hurt the water quality and will quickly end up in the filter.
I'm quite lazy with nets.. I leave them laying about and just give them a little tap before using them to get any dog hairs off! I don't think its something which is worth worrying about unless you have many tanks, use the same nets between them and there's disease or something in one tank which you don't want to spread.