Take a piece of parchment paper and lay it out on a baking tray or sheet. This will stop the parchment paper from collapsing if you need to move it later.
Using a pastry brush, paint your sourdough starter all over the parchment paper ensuring it's not too thick. You can use a spoon if you don't have a pastry brush, but try to get it nice and thin.
Leave the sourdough starter in a warm place with good air flow to completely dry out. You don't want it to blow around, so make sure that it's not right under an air vent.
Check the starter periodically and break up any pieces that aren't drying out completely. Breaking them into smaller pieces will help to ensure the starter dries out in a timely manner.
Once the starter has completely dried out (usually takes up to 24 hours) you can break it up into pieces and place into a clean and dry glass jar.
Store the dried sourdough starter in a cool, dark place. It will last indefinitely as long as it doesn't touch any moisture.