Set in an alternate 2010s/2020s, floating nuclear power station are now common as they can provide power to any city it is docked at (think along the lines of this real world example. or this powership) As these power stations are now widespread, pirates and terrorists find these atomic energy barges an attractive target for sending a message to the world (usual demands- money, power, etc.) What would be the repercussions if one of these floating power stations were to go into meltdown? Would the seas become radioactive as a result?
It depends. We typically call any incident in which part of the core of a reactor melts a "meltdown". But not all meltdowns are equal. At the worst case, you have something like Chernobyl, where the core melts right through the containment vessel and into the wider world. In this situation, huge amounts of radioactive material will be released into the ocean and will spread by the currents. You will also get a great deal of steam, which will aerosolize some of the radioactive material and allow it to be spread by the wind.
On the other end of the spectrum, you have a small, partial meltdown in which no core material escapes its containment vessel. This would have no appreciable effect on the ocean. It might not even be detectable outside the facility.
One thing I'd like to point out is that most materials don't become radioactive when exposed to radiation. The water would not become radioactive itself, it would have radioactive material dissolved in it. Still very dangerous, but it means a cleanup program is possible, at least in theory.
As a commenter mentions, this would be similar to Fukushima. In fact it would almost be identical. Despite improvements in safety measures, a good terrorist attack would likely get around many of them. The immediate city around it would be come radioactive enough that it would be unlivable. On top of that, a large number of people would be exposed to dangerous levels of radiation. Very few would cause death in a short time frame though. The radiation dose from a nuclear reactor melt down can cause serious damage for someone in their future. Not to mention that there'd be a small dead zone in the ocean for a while, but it should get dispersed within a few weeks assuming the radiation source is consumed. As for the seas becoming radioactive, that may be a bit of a stretch.
Edit: it seems I didn't take into account that there is a large variation in how bad it can be, but I'll leave this here
It depends when they were made and by whom.
Old power plants when they went into meltdown were bad (Chernobyl) and the likelihood was higher.
Newer ones like Fukushima were not so bad.
However, if they are recently made then the likelihood of meltdown is tiny and when they do nothing much happens. Modern nuclear stations are so well designed (ones in sensible countries (UK, USA, France, Germany)) that they shut down safely.