While cutting the cord after your baby is born might be a fairly safe procedure for dads here in this country, in developing nations it's not that simple. A San Antonio biotech company has apparently thought this through and created the Joey™ Umbilical Safety System (I guess baby koalas are also called "joeys"). It cuts and clamps the cord in onesanitary procedure, while preventing cross-contamination of potentially infected cord blood. The koala-faced clamp then stays on the child until the stump falls off. It will even leave some extra cord for testing, or if you're Korean maybe for gold plating and hanging on your wall.
While it's depressing to think of HIV positive babies being born and potentially infecting health care workers, in some countries I guess it's more common than we would want to believe. Hopefully this will help.