DS just had a bath every evening (which in itself can be frowned upon for a newborn but worked for us) - didn't do anything special (not even boiled water), just took care when changing nappies and if fell off quite normally leaving a cute innie.
You are advised not to clean it with anything these days (unless specificially told to because of an infection or whatever). It slows down it drying out and falling off and makes it more likely to become gunky. You just leave it well alone until it falls off. Then clean the area with cooled boiled water. You are also advised not to bath them until it comes off too.
I live in South Africa where there's a high risk of neonatal tetanus so Surgical spirits is a must for cord care. the other option is a homoepathically prepared powder nmade by weleda calles wecesin - which is specifically for cord care, weeping wounds