Starting Point for the Zanskar is usually Padum or Ringdum. There isn't much of White Water there, so a direct drive to the Karsha Monastery and putting in at the bridge below Karsha would be possibly best for a self supported kayak trip.
To get to Padum, it is best to drive via bus or jeep to Kargil and then get a bus or truck to Padum. If u leave Kargil at around 0500 pm, u would make it to Padum in one day. Which depends on the driver itself and how much he wants to be in Padum. The Fare for the Truck is about the same as the bus but, the bus will definitely take 2 days, is crowded and will take extra charge for your boats.
There is some extremely serious boating on the upper reaches of the Zanskar where the Drung Durung Glacier comes in. Also are there some unexplored side creeks coming in at the Padum area worthwhile to give a try.