1. They tried that and it didn't work. The aliens just sent more Kaiju.
2. Irradiating the ocean is not a good idea.
3. The Kaiju will just keep coming. Even worse they are coming with increased frequency and number.
No. In the film they stated they tried to send a nuclear bomb to the other side (the Kaiju's dimension) - and that never worked as the portal was thought to be 'one way' (until the end of the film of course).
They never stated they tried just wringing the portal area on our side with nuclear bombs that detonated as the kaiju finished appearing.
As for irradiating the ocean - the U.S. did that for years in the 1950ies; and they found the ocean itself dispersed the radiation so well, the waterbourne environmental impact was negligible. (The effect on the land portions of the Bikini Atoll were greatly affected; but again, we're talking underwater.