If you have multiple aquifer layers there is actually a way to get drop a 3x3 hole through them rather than the kind of pit mine you'd probably resort to. It's a bit tricky but the idea is that when you channel down into the next aquifer tile it will absorb any water from the channel above it. You've got to be very careful with that though.
As for piercing regular aquifers: Each tile you mine through is composed of a wall section and a floor section. Digging takes out the wall obviously and channeling takes out the floor tile along with a wall below it. When you place up stairs they sit in the wall position while down stairs replace the floor with something that can be walked through, provided there is something suitable below. Stairs do support structures.
Floors are very thin and won't take up any space if dropped. Rather it takes walls to accomplish this. In order to pierce an aquifer like this you must have at least 2 layers above the aquifer. If there are more layers of aquifer you will need a much, much more complex plan. More on that later.
So then, about actually rigging your plug. You can either place a support above or below the wall chunk you are going to drop. If above you'll need to make a wooden arm or something out of constructed floors or walls. Also getting on and off of the plug may seem like an issue but diagonal tiles do not hold things up so you can just place a floor at the corner. If below you only need to suffer losing the materials for the support and mechanism.
1. So then, if you want a 3x3 hole in the aquifer you will need a 5x5 plug. Mining away the tiles of the plug is obviously not going to do you any good so you need an insulating ring. Also you will want to mine out a 7x7 area on the layer just above the aquifer, as the plug cannot be held up by floors or any of that next to it.
2. Channel out the tiles in the aquifer for your plug. Best to just do one line at a time if you don't want the miners getting stuck or complaining.
3. Construct some floors over the pool if your support will be under it. Place your support and link it to a lever.
4. Channel out the tiles around your plug so that only the support holds it up.
5. Pull the lever.
6. Either Dig out some more space for the water this knocked up to spread out of pump it away.
7. Dig your 3x3 stairway down through the middle of the plug being careful not to ruin any of the insulating ring that keeps you dry.
Multiple aquifer layers can still be defeated by the same number of dry soil layers but will definitely require some pumping. The idea is to mine out a much larger area so that you can just drop rings of soil into the aquifer. So mine out large area, channel out first ring into aquifer. Build support for the ring as well as support for the later stage plug inside the ring. Drop the ring, channel out the aquifer tiles inside the ring. Pump the water out. Step down and repeat with a smaller ring.
However if you'd rather just not deal with aquifers it is advisable to generate worlds with less ocean. Obviously this means not making island regions, and a lack of edge oceans also has the advantage of making caverns much closer to their defined depths in custom world generation.
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Edit: I found the ring aquifer pierce thing illustrated.
http://www.bay12forums.com/smf/index.php?topic=64625.0-
Huh, you think I'd get a thank you or something.