Somebody did this. Before the experiment, they expected a background level of Cesium 137 of between 1 and 2 bq/M^3 (caused by 1950s nuclear testing, which released a ton of it). They found actual levels between 1 and 2 bq/M^3 across the Pacific with some small variation above and below: http://ourradioactiveocean.org/results.html

Think about it this way. Worst-case estimates are that 2-4 kilos of Cesium 137 were released. If it all ends up in the ocean, the total Pacific Ocean weighs on the order of 638,000,000,000,000,000,000 kilos. The total new Cesium 137 is: 0.000000000000000000627% of the ocean. At a 30 year half-life, that's one decay event per 70 liters per hour.

Not going to happen, nor is there much reason to do so. For one, the NaI detectors are relatively inexpensive, but still you're talking about $10K each minimum for such a setup. There are other detectors that would work for this (EJ-309 for instance, you need a detector capable of energy spectroscopy), but they're even more expensive and often extremely toxic. What I was describing is a setup to measure contamination of the water while it's still at the plant.

These sorts of statistics are monitored and the data is available ( http://www.biogeosciences.net/10/6045/2013/bg-10-6045-2013.p... ), but the contamination levels are so low that you have to use a very different method. Basically they go out and collect a bunch of sea water (linked report uses 100 liters), then filter it through a microfilter and the filter, which traps all of the cesium, is measured back in a laboratory using a very sensitive HPGe detector (which cost $100K+ and require cryogenic cooling at all times).

The measured activity levels in that report were in the range of about 1-15 Bq/m^3 of sea water. That's 1-15 atoms disintegrating per second (there's ~10^29 atoms in a cubic meter of water). Those levels are far too low to reasonably measure at sea, end of discussion. Even the highest observed dispersal of about 140Bq/m^3 (which was actually Cs-134 and was in a small area immediately after the disaster) would be very tough to measure in real time like you suggest.

Much more practical is to monitor the rate of dispersal at the outlet (i.e., the pipes pumping water into the ocean at Fukushima) and simulate the fluid dynamics to calculate the dispersal. These results you then validate by comparing to spot measurements taken in the manner described in the linked paper. In fact, this is again exactly what they do. This work is widely published and openly available, but not well known to the general public because it's rather technical. My own research work is on a related subject, but different application (finding the position of a concentrated radioactive source in an urban environment).