Water pretty much can't get radioactive. (For more than a few minutes.)
The longest radioactive oxygen you can make in a reactor has a half life of 26 seconds.
For hydrogen, deuterium is not radioactive, and tritium can't be made in a light water reactor.
So water effectively can't get radioactive.
When you say the contaminated water can be kept in the containment, does that mean that they are basically flooding the containment with seawater? My impression was that the there is still a need for cooling (i.e. cool water in, hot water out) in case of a total melt down.
If they could actually circulate water they wouldn't be in this situation.
The worst sodium has a half life of 15 hours.
The longest chlorine has a half life of 360,000 years, but is barely radioactive. The rest are all under an hour.
There's other stuff in sea water though. But not in large quantities.
The water is being boiled away, not circulated with pumps.