Baby's belly button bleeding

My sons Cord fell off last week (at 8 days old) every now and then he still has a little bleeding. The public health nurse said it was fine to just keep cleaning it with water which I've been doing. It's not bleeding a lot or alarming but it's getting dried blood on all of his clothes. Just wondering if anyone has figured out anything to help it stop. I read about putting alcohol on it  but I'm hesitant to do that I don't want to hurt him. Today I taped a cotton ball over it to protect his clothes...any tips ?!?