At first I didn't see the problem, "Ok, so it's a bit messy, but it all drains off into the ocean at the end." And then I looked down a level "Oh wait, that isn't the ocean."
If you want to make it looks nice, you could carve a river down at the bottom. Unfortunately you can't remove ground from the edge of the map, but you can do this: Carve down several layers deep near the edge of the map until you get down into rock, then smooth the rock over, then turn it into fortifications. Then channel a river from where your waterfall hits the ground to the edge of the map there, and it should flow nicely away. I suggest doing the same where the waterfall is falling down, channel the areas down a layer, so the water is always a layer below the surrounding rock, and it should flow more in a straight path if that's what you want.
Although doing that might be a bit difficult with the river still flowing, so to do something about that, I suggest constructing a screw pump, or a few of them, along where your diverted river is, and power them (waterwheels or windmills recommended, you can use dwarves to pump them directly if you're in a hurry, but they can be unreliable if they run off while you're in the middle of doing something in the channel). And then you can bring people into the dry channel and build a wall, or build floodgates, or a drawbridge that can raise, hook them up to a lever, and then you'll be able to control if your side passage is open or closed. At which point you shouldn't need the pumps or power anymore, you can keep them there, or disassemble them for parts, or put them to other used.