How long should a minivan transmission last?

As long as you keep up with the maintenance, depending on manufacturer, it should last well over 200 thousand miles. I've seen Asian imports (Toyota's and Honda's mainly) with up to and including 500 thousand miles with the original engine and transmission.

Regular maintenance includes and is not limited to, Fluid and Filter changes, Transmission flushes, Transmission pan gaskets(if leaking) and regular checkups.