What's the easiest way to build down into water?

Water is notoriously difficult to remove in Terraria, placing blocks over it will not work as you have already tested. Here are the few methods I know of from personal experience:

• For small pools, buckets are a simple but tedious method of transporting the water elsewhere.
• Large pools can be drained into below caves or sent directly down to The Underworld where it will form obsidian on contact with lava and/or evaporate.
• There are other ways to remove water by utilising exploits and bugs but for the sake of avoiding cheats, I won't list them here.

Alternatively, you could keep a small amount of water in your base for use as a crafting station or fishing spot. I suggest the easiest solution would be to use a hellevator to easily drain water into The Underworld.

On the other hand, if you'd like to construct a building inside a large pool of water, you can use either of the previous methods to great effect by building up your walls to hold back the water you'd like to keep before draining the interior.

Actually, I did a test and I did find a way to "remove" water from where you want to build.

In my case, I created a platform over the water, and then I filled rock in as far as I could reach. Now, typically if you dig out the rock, your new, empty space will fill back up with water.

However, after creating the blocked, rock region, I stopped the iOS app and restarted it. After the restart, the rock seemed to exist without any other settings persisting that would force that area where the rock was residing to contain water. So, I dug out the rock, leaving only the sides holding back the water, and no water remained in the enclosure.

I guess this is using a bug/exploit, but it is allowing me to build through water. It's not perfect, but it does work.

The easiest way, in my opinion, is to simply dig a shaft out the bottom of the water until you run into a cave, and let the water drain into the cave.

Depending on how far into the game you already are, there are ways to make it easier to dig through or drain water to make way for whatever you want to do.

Flipper

With enough luck and courage this can be obtained right from the start of the game by running to the ocean and doing a suicide/log out dive and one of the chests might just have it. Optionally large pools of water around might also have water chests that contains this if you search around.

With Flippers you can dig downwards till you are almost out of breath and just swim to the surface and catch a breath. Repeat that process a few times and eventually you'll dig into a large enough cavern to drain those water away.

Flippers can also combined with Diving Helmet later to make Diving Gear which also increases the duration you can stay under water making it even easier.

Gills Potion

This is also obtainable very early into the game, if you are lucky, some random chests or pots around might just supply you with some. If not so, you can combine Coral obtainable by diving slightly into the ocean with Waterleaf which can be found in the dessert together with Bottled Water which can be found around or by filling Empty Bottles near pools of water.

Application of Gills potion is even easier, once used just spend the next 2 minutes digging down and draining the water away from where you do not want it.

Neptune's Shell

If you are already well into Hard Mode, crafting a Neptune's shell effectively combines both the previous items I suggested together and allows you to freely move indefinitely in water.

If slowly digging your way through is not what you have in mind, I can also suggest

Dynamite

Sometimes you might find it hard to drop the dynamite exactly where you want it. You can combine it into an Explosives. But needless to say this will result in almost every tile in the area being removed - except for tiles immune to explosives. You'll have to build back the entire area from scratch after using dynamite.