A cotton swab is usually the easiest way to clean the umbilical cord. This is because the cotton swab can reach further into the belly button than a cotton ball can. When you're cleaning the umbilical cord hold the cotton swab in one hand and with your other hand either press down on the sides of the umbilical cord stump or excess skin. This allows greater access to the inside of the cord, where it will stay "wet" longer.
You should call your baby's doctor or pediatrician if there is a foul odor coming from the area of the umbilical cord, if there is redness around around the cord or if your baby is running a fever.
Photo © Robin Elise Weiss