Two ways this happens:
- Water can become radioactive via neutron activation.
As neutrons are released they fly around and can get captured by other atoms (such as water). When that happens the atom wants to decay back to its original state. That decay is radiation.
In water, from what I recall, the process is relatively short lived (a day or two?). For this reason when the Fukushima reactor was venting steam there was not a lot of worry. It’d float out to sea and decay before it met another population.
Note in reactors they purify the water. One of the issues at Fukushima when they dumped sea water in was sea water has all sorts of stuff in it. Pure water does not get activated easily. Sea water can have all sorts of things that, if activated by neutrons, can be dangerous.
- Radioactive junk can fly around and end up in water.
In this case other materials, such as Cesium, can land in water. The Cesium can be radioactive and you can drink it.
This is the bigger problem.