Silk is a beautiful fabric and a silk table runner is a great addition to any table. Silk requires special care to maintain. If the label says to dry clean only, then that is the best thing to do.
Step 1 - Wash Gently
Test a little bit to make sure the color won't run when wet. Hand wash, or machine wash on the gentle cycle in lukewarm water. Wash each color separately. Use just a little gentle soap or shampoo. You may want to wash silk items in a cloth bag or pillowcase. Rinse once, then add 1/4 cup of white vinegar and rinse again. This gets rid of soap residue. If there are stains on silk, treat them as soon as you can.
Step 2 - Dry Carefully
Dry in a dryer only on the "fluff" setting with no heat. Otherwise, roll the fabric in a towel to get the water out, then lay the runner out flat on towels to dry. It should not be in direct sun. You can iron on low heat, but only on the underside of the fabric.
Step 3 - Store Well
Silk should be stored away from light and heat, and protected from insects. Moth repellent can be put into the bags.