It is a workable idea. We have net fabric available, and net designs that would keep anything larfer than a golf ball out of the net intake, but still be able to remove tons of the stuff. In water it tends to seperate into layers by density too, so you would end up with a fairly consistant, relitivly clean take of recyclable plastic. The problem is twofold. Most of the plastic is going to be an ABS or PVC derivitive. ABS can be purchased clean, ready to go in pellet form for about $0.50 a pound retial, PVC even less. Lets say you get a ship to the garbage patch, set it to scooping, and you harvest 3 tons an hour for a 12 hour shift (which is about average for a midsize fish factory ship fish haul from a seine net) you're looking at around theoretically $36,000 worth of plastic per day. But you have to subtract labor, fuel costs, ship cost, equipment costs, etc from that figure, and thats not even taking into account the likely reqirement to further sort and process the stuff after you collect it. By the time you factor that all in, it would be more worth the money to go salmon or crab fishing with the same size boat, or just haul cargo back and forth.
The other big problem is that the pile of junk is in international waters, which means no government is in a hurry to offer any tax incentives for people to go clean it up. I'm actually suprised one of the bigger corporations out there hasn't launched a small fleet of clean up ships anyways though just for the public relations gain, but there are sources closer to home that are easier to aquire for higher profit margins that haven't been exhausted yet. If you have the $1,000,000 to throw at a plastic recycling plant in one of the areas that isn't already covered by recycling contracts, you can pretty much make your money back in a year and then double it on domestic recycling corporate incentives.