Daylilies are among the hardiest plants around. In areas with mild winters, they stay green all year round, and in the spring and summer sprout beautiful yellow to yellow-orange flowers. To encourage blooming, it is recommended you "deadhead" your daylilies so new blooms can develop. Dead flowers also sap energy and nutrients the daylily could otherwise use to produce new blooms.
Go through the daylilies and snap off any old blooms with your fingers, making sure you get the seed head off as well. The seed head is at the very base of the flower. If you don't remove the seed head, it will eventually dry and split open, but generally the seeds are sterile--so it's much better to snip off the spent flower so a new one can develop.
If there are no more blooms on a stalk, take a pair of clippers and snip it off at the very bottom, about an inch from its base. This will "trick" the flower into thinking it has lost a living stalk, so it will send up a new one. And new stalks mean new flowers.
Remove any stalks that are already brown. Chances are you are too late for a new stalk to develop, but the less dead plant matter the better.