I dam them and use the enormous water supply for, well, whatever I can think of
how should I go about damming it?
I'm not sure pumps would be effective...
1: Construct a 2xhoweverwideyourriveris floor suspended above the river, This need to be high enough that it can be dropped into the floor of the river and span it completely. Usually 1 z level above the riverbanks, supported on one end with a support rigged to a lever. If you really want you can build this last to make sure no random lever pullers flood your fort.
2: Directly below the river dig out a chamber and some associated piping in order to give you 1)a controllable water source and 2) a controllable drain for the river either off the map edge(via fortifications or into caverns), or into an aquifer. The piping can all be 1 wide, as long as there is a sufficient drain at the end. Use floodgates or bridges for the controls. Dont build anything directly under the bridge suspended above the river or it will get exploded in the next step.
4: After getting all the piping dug out and prepared (and properly secured for cavern drains) collapse the 2xwhatever bridge through the floor of the river. The river will now drain out REALLY fast, yet rape your FPS to the point where your computer might be thinking DF crashed, just wait it out. After DF finishes its surprise intercourse with your CPU the river should be draining into the hole across the bottom of your river now and one half should be drying out eventually.
5: Once the one side is dry, go down there and either build a wall or set up raising bridges or floodgates or whatever you want to use, once they are built close off the drain below and viola, your river is dammed.
Aquifers can screw with this one, in the case of aquifers I use the magma method.