All the kids with Geostigma - and, presumably, Rufus too - were infected by the Lifestream itself. After Jenova was defeated by the heroes in FFVII, her essence dispersed into the Lifestream. When the Lifestream came forth to destroy Meteor, Jenova's tainted spirit energy was carried with it, infecting anyone who came into contact with the tainted energy. Midgar was where the Lifestream converged, so that's where a lot of victims came from. Kids seem to be especially vulnerable, perhaps something to do with the fact that they're still growing.
(SPOILER)Reeve's mother was one of the first people to die from Geostigma, dying a few hours after her exposure to large amounts of the contaminated energy. Others, like Denzel, began showing signs more slowly.

Virtually all of this info is from the novella, 'On the way to a Smile: Case of Denzel', and from the movie itself.

Kadaj is actually kinda telling the truth when he tells the kids that he and they are brethren. The kids are carrying fragments of Jenova's spiritual essence that came from the Lifestream, while he and his brothers are fragments of Sephiroth's spirit that didn't disperse.