Elements in the cooling water used in Japan are bound to be transmuted to radioactive compounds, which could be a problem. The process is called induced radioactivity.
Here’s an example of how an element in cooling water can be converted to a radioactive form in a nuclear reactor, from a supplier of cooling water additives:
natural Zinc contains 48% Zinc-64 and as the cooling water is subjected to continual neutron bombardment this isotope is activated to radioactive Zinc-65. This isotope is a strong radiation emitter with a long half-life and thus contributes greatly to the storage time and hazard of waste cooling water. [my emphasis]
Now, the concentration of zinc in seawater is low (about 5 parts per billion in one source), so radioisotopes of zinc would not be formed in large amounts. However, the concentration of elements in seawater are far larger. For example, sodium is at 10,800 parts per million (cf the same source).
I’m not clear on what is happening to the cooling water being sprayed on the reactors in Japan. If it runs off, then it is likely to contain radioisotopes, and it will be important to think about this issue once things are under control. If it comes off as steam, then that steam is likely to have within it some level of radioisotopes formed by induced radioactivity.