Saving the corals might actually be the easy part: It wouldn't be a fun job, unless you are a real saltwater aquarium masochist; but taking 'cuttings' and propagating them in captivity is reasonably well understood, at least for the ones that have historically merited the attention. Even if you can't modify them to make them more durable, there is lots of ocean currently too cold for a given coral ecosystem that, if warmed, will become a viable location for transplants from the areas that are becoming too warm(the current cold-water reefs, which do exist; but don't get the attention of the tropical ones, may be pretty screwed, Rost reef, in the frigid waters off Norway, will be pining for the fjords).
However, much of the charm of coral reefs is the amount of ecosystem that they support. All sorts of weird stuff, the stuff that eats it, and so on to the top of the food chain. Transporting that, or convincing it to swim in the right direction, will be a much greater challenge. Plus, unlike the fecund swarms of tiny organisms, where 'genetic diversity' fits in a medium fish tank, and could probably re-mutate from a monoclonal strain in a matter of decades; larger organisms lose genetic diversity much more easily as individuals die, and don't recover nearly as easily. Some of the larger fish, say, will be will be a pretty inbred and sorry lot(exquisitely vulnerable to disease, as monocultures always are), if most of them die and a new population is seeded from a few transplants.
If there were some sort of payoff in it, we could probably have 'coral farms' up and running in short order; but they'd have roughly the same resemblance to natural reefs that tree farms producing papermill feedstock do to mature forests, or alfalfa fields do to prairies.