Until baby's cord falls off, avoid getting it wet. If it gets dirty, clean with a baby wipe or rubbing alcohol. Instead of full baths, try sponge baths. Gather a basin of warm water and clean washcloth so you can keep one hand on the baby at all times. Lay a soft towel on a clear, stable surface for baby to lie on. Keep them warm by leaving them wrapped in a towel, only uncovering the part of baby you're washing. When the cord does fall off, there may be slight oozing of blood. Don't worry about this -- just keep the area clean with soap and water. Talk to your doctor if you see pus or redness of the skin around the stump area or if you notice a bad odor.