It seems like a kiva would be perfectly suited for gathering rainwater and not being able to drain that water out. Even though some Great Kivas had ledges raised up at the circumference above the surrounding ground, I've read that most Chaco-style kivas were flush with the ground. Also, in the Western U.S., I can't imagine roofs had fronds or anything like that to do a really effective job at being watertight. Did they use skins or something?
This form of architecture typically occurs in places with fewer than 10 inches of rain per year, so any single storm was not likely to flood even an open pit. The roofing was sufficient to slough off most of whatever rain fell.
The famed desert flash floods occur when a single storm dumps nearly all the precipitation for a given year - all in a matter of minutes, which is not uncommon. The various drainage channels of the desert gather the water into larger channels at valley floors, causing dangerous floods of water. Native Americans were sufficiently aware of the environment not to be so foolish as to place a kiva in these places - or if they did, they were awarded a Darwin award and the genetic pool moved on.