Really, water doesn't become radiated. I forget the specific region for it, but water doesn't absorb and hold radiation the way you're thinking it would or should. In fact, water's a good insulator for radiation. You could technically swim in a storage pond with radiation at the bottom and not suffer any effects until you get within arms reach of the material at the bottom.

Of course the people storing it wouldn't let you for safety reasons, don't want ass holes proving they can and swimming to the bottom of their deep storage wells to show they can. If anything the biggest effect I'm aware of that radioactive materials has on water is in thermal radiation, where water will boil off of active nuclear fuel for power-plants making it necessary for some measures to keep the insulating, cooling water in.

Really, the risk of "irradiated" water is contaminated particles in the water. But spread over time in the flow of the water and settling somewhere far away would dilute the danger of radiation sickness to the point it doesn't simply matter. Unless you have Chernobyl upriver from you which would require filtration to catch the radiated particles flowing in from up-stream and this being continually refreshed by being continually exposed to a source of radiation. But you should really ask Kiev about their water in the end; their down-river from Chernobyl.