First. Dumping all the nuclear waste from the process of nuclear power into the ocean would not be a proper answer. Of course we care about the "things" that live in our oceans. The radiation would not only poison the life in the oceans, but through the food chain would continually poison all creatures that consume those creatures and all the creatures that consume those creatures and so on. Humans are ultimately at the top, and we would lose one of the largest food sources in the world. This is just an immediate effect. The half-life of the isotopes from the byproduct of nuclear energy varies, but the effects would persist for decades (I am not a nuclear engineer or physicist, but this is just basic high school chem, bio and physics).Not to mention dumping radioactive material in the ocean would not result in it sinking, but just some sort of radiation soup of the ocean that would spread via the Coreolis Effect and the ocean currents, reaching America, Australia, Oceania, Asia, Africa, and in smaller effects, Europe, likely. Levels of radiation have been detected on the west coast of the United States due to the Fukushima Daichi disaster. That was just a single, isolated event. To continuously dump the amount of nuclear waste from the proposed massive nuclear power system on the Polynesian and Micronesian islands would be terrible.To say that dumping the waste in the ocean seems that it would be analogous to saying that dumping pollutants into the atmosphere doesn't affect us because we're on land. It would do immense harm to the ecosystem and would cause a massive collapse of the environment, likely. Not to mention the difficulty of transporting energy across the oceans to any of the continents. It would require an extreme amount of resources if you plan to send energy through electricity, because of the degredation of current and resistivity of wires, etc. Power plants are best organized as they are, I would guess, because they are more concentrated to areas where there is a greater demand for energy and a ready and easy supply of the resources to power it. The Pacific Islands have none of these things.
As for the reason for opposition to nuclear power, it could be driven by political and economic support for the all-mighty coal and gas power companies, but it also because of a more-likely-than-not "bad rap" for nuclear, with Chernobyl, Three Mile Island and Fukushima Daichi. Public awareness of the possibilities of safe and extremely efficient energy would be helpful in their support. The original two disasters, on a basic level, were caused by a lack of knowledge on the technology, and all three because of insufficient safety regulations. If the regulations were more strictly enforced, then it might be a different story.
I have not heard of the previously mentioned reference to cancer incidents around nuclear power plants, but a similar public health violation can be thought of in coal and gas power plants, along with hydraulic fracturing. All of these problems could be fixed by tight regulation on emissions by the government in the respective areas such that it is both clean and profitable. Hopefully.
EDIT: Typo