The cord stump does not need any special care. It is OK for the cord to be wet with bathing or urine from a wet nappy. It is also all right for the nappy to sit over the cord (although many parents will fold cloth nappies underneath it to stop it from pulling, or alternatively, turn the top of disposable nappies down). If the cord stump becomes soiled with poo, simply wash with clean, warm water and pat the area dry.
When the cord becomes gooey and sticky (after about 4 to 6 days), you can clean the base of the cord with a clean cotton bud. Some parents will use plain (possibly warmed) water with the end of a cotton bud. This is all right to do occasionally (say once a day), but it may take longer for the cord to separate if continually moistened. The cord itself has no nerves, but the belly button can be a little sensitive. The baby may cry if the water is cold. Remember, if using warmed water in a bowl or container to wash dirty bottoms, do not use this same water (or container) to clean the cord, otherwise you may introduce infection.
Information sources
Airede AI. Pathogens in neonatal omphalitis. Journal of Tropical Pediatrics 1992 Jun; 38(3): 129-31.
Enkin M, Keirse MJNC, Neilson J, Crowther C, Duley L, Hodnett E, Hofmeyr J. A guide to effective care in pregnancy and childbirth. 3rd Ed. 2000, Oxford University Press, Oxford.
Zupan J, Garner P. Topical umbilical cord care at birth (Cochrane Review). In: The Cochrane Library, Issue 4, 2001. Oxford: Update Software.