The important feature is to use a brush that is as dry as possible. Rinse it under the tap when it gets dirty, but don’t rinse the truffle itself.
It’s always tempting to just run the truffle under the tap to remove the water, but please resist. Truffles actually have a very delicate water balance, and you don’t want to interfere with it. Example: if you store a truffle in a Tupperware container in the refrigerator overnight, you will find the top covered with condensation the next morning, something that doesn't happen with an ordinary mushroom.
When storing, put a piece of paper towel inside the sealed Tupperware container to absorb any excess moisture.
If you are using truffles to make an omelette or risotto, store them in Tupperware with the uncracked eggs or rice overnight. The next day, you will find that the aroma and flavour of the truffle have permeated the rice and even the shell of the eggs!