A lot of things would likely happen. I'm not an oceanographer, so I'm just making guesses here, and am probably missing things from this list I've come up with. Nature tends to be in a pretty fine balance, and significantly changing parameters like the strength of the ocean current can have far-reaching effects.
Mentioned already in the comments, you'd see rising water levels from the faster-melting ice caps, which by itself would wreak havoc. I imagine you'd also get temperature changes in places, as warm and cool zones end up mixing more, which could pose survival problems for certain species. Not to mention that pH levels would probably get thrown out of whack even more than they already are (which may be one of the reasons seastars have been melting lately). If the problem wasn't addressed in time, you'd end up with whole ecosystems collapsing as keystone species go extinct. For example, coral could start dying off even faster than it already is, which would have a serious impact on everyone. Soil erosion would pick up along coastal regions, too, which would have a negative impact as well.
You might also see a shift in weather patterns, since the ocean tends to be a massive heatsink. Maybe you'd see more frequent El Niño seasons, for example. This would also be really bad for the ocean's ecosystems, and thus really bad for our ability to keep everyone properly fed and housed. I suspect you'd eventually see political upheavals.