Here's something I concocted that's compact and only requires one dwarf-powered pump. It drains the aquifer back into itself, and doesn't get it all over the surface:
2x4 double-slit method
Dig a 2x2 staircase until you reach damp stone, don't dig it into the aquifer soil! Channel out 2 slits to either side. Remove 2 up staircases between the slits, floor over them and build the pump there:
Man the pump. It will drain one slit into the other. Start digging out and walling off the quadrant being drained. Don't dig them all out at once; the more room there is for the water to push your construction worker around, the longer it will take to finish the wall. The corner towards the center needs a wall, the outer corner does not.
Dismantle the pump and re-install it facing the other direction. Repeat the mirror image of the above step.
Dismantle the pump and its flooring, replacing them with constructed staircases. Remove the other staircases, floor over and install the pump.
Start walling off the third quadrant, but leave the block of aquifer near the center. This is important!
The block will act as a sink for the water pumped out of the last quadrant. Add a wall in the slit, adjacent to where the water is being drawn out.
Almost done! Dismantle the pump and re-install it facing the other direction. The pump may spread the water a bit on its level,
but it isn't dangerous.
(2012 Update: knocking a dwarf down a z-level can be dangerous now. Building a single wall directly on top of the wall built in the slit should stop the splashing.) Wall off the last quadrant. With this slit dry, you can remove the center wall diagonal to the aquifer block. The aquifer will not add water through this diagonal, but your miner can come in and destroy the block. Since we added another wall to the other slit, there should only be a total of 7-9 units of water in the opened area, which will evaporate.
Don't remove all
the internal walls until you have stair access or dwarves might get stuck there, unable to use the ramps. Dig down stairs to reveal the layer below, then build up/down stairs on top of them for access. Building the stairs will remove any 1/1 water on the tile, and render the layer below dry if it is not an aquifer.
You can repeat this as many times as necessary to get past a multi-layer aquifer.
Edit: I forgot if water will go in
to an aquifer on a diagonal. If it does, it isn't necessary to leave the block there.
Edit 2: No, it won't. You either need the block, or a channel into an aquifer layer below in its place (or better yet in one of the slit tiles after it's walled off).
Edit 3: A faster way to get to the bottom aquifer level
You can power through every aquifer level except the last much more quickly by using this modified method, then switch to the above method for the last layer.
The setup is similar, but instead of channeling, dig down staircases for the slits, then up/down staircases below them that will fill with water. (Multiple levels shown, color-coded for convenience.)
Install the pump as before, but annoyingly you must remove the stairs from its input tile (on its level) for it to work. Pause/one-step until there is no water in the tile the pump is drawing from. If the layer revealed below (yellow) this is not an aquifer it should be dry during this frame (not tested), and you should switch to the original version. More on the conversion below. Similarly, if there is an aquifer but it looks like one of the slits won't drain into an aquifer based on the revealed material (yellow) you may want to switch methods and consider your options for dealing with the non-aquifer rock.
With the pump active, dig up/down staircases 1 level deeper into the aquifer. This will act as a drain.
Re-construct the pump facing the other direction, remove the blocking staircase and repeat.
Stop pumping. You don't need the pump any more for this level.The water will now simply fall down the staircases. If you build these walls in this order you may not get a single cancellation!
Carve out the center with staircases and mining (Don't carve staircases in the layer below!). The first of the 4 walls will push your builder all over, but it will get easier.
You're ready to conquer the next level! When you reach the last aquifer level, you can switch to the above method by building ramps in the slits. You can build the ramp directly on the carved up/down staircase, but remember dwarves can't use the ramp if there is a staircase directly above it. You may have to remove a ramp to build a wall in the slit as part of the above method.
Edit 4: Check out Panando's accelerated version on page 5.