They missed on the good chance to climb all over Stanton. Like the Pirates did with McCutchen ... the time to pounce is early, give him more money than the rookie scale would have, and try to buy out his arb raises and (maybe) a year of FA. It will be hard to keep him. Dodgers far and away the most likely match though.
One of the quirks about the Marlins is that while the ownership is one of the worst in the game and the management at the (Owner-Survivor Contestant) level is shaky, the baseball operations guys have largely been pretty good. It is a horrible market - so extending Stanton on its own is difficult. To use econ jargon, the marginal value of a win there is really low (comparatively in Boston it is excellent).
Historically in these sorts of dumps, the Marlins have wanted power arms. This is the organization that grew Brad Penny, Josh Beckett, AJ Burnett, Anibal Sanchez (of course, they got him for Josh Beckett - but describes the sort they target in trade), Jose Fernandez. The Red Sox have Cuban guys who can help the Miami marketing sure - but the system, while deep with pitching, does not seem to have the fireballer sorts that the Marlins have long chased.