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.

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.

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.

The easiest way, once you have the item, is to use a Super-Absorbent Sponge, obtained from the Angler's quests in hard mode.

Using the sponge on a square simply destroys any water it contains. With that, you can wall off an underwater area and empty it without any trouble.

When working under water, I use a couple tricks to make life easier.

Cutting breathing stations into the wall

X = blocks

~ = Water

XXXXX~~~~~
X   X~~~~~
X~~~~~~~~~
X~~~~~~~~~
XX~~~~~~~~
XXXXX~~~~~

Of course this technique is not particularly useful on in sandy spots like the ocean floor. In this situation build a sealed phone booth, dig a row of sand from the floor to create an air bubble at the top, then open a door.

When a hellivator needs to go through Water:

1. Minimize the water that needs handling by making dams on either side of the shaft. Sand or Silt can be used to minimize wasted time. Once your water problem is handled, you can construct more sturdy dams by creating a 2nd column of sand, digging out the 1st, putting stone where the 1st wall was, and finally digging out the 2nd column.

2. When digging down in a water filled shaft, drop a glow stick, then build water collecting side shafts.

Example: (presuming a pickaxe reach of 4 blocks)

XXXXX  X
X      X
X~~~X  X
X~~~X  X
X~~~X  X
X~~~X~~X
XXXXX~~X
X~~~~~~X
X~~~X~~X
X~~~X~~X

When the water level drops below the intake of the side shaft, start a new one. You shouldn't need too many before you're water free again. Feel free to do this on both sides of your main shaft if you don't require a breathing station.

My experience has shown that generally in terraria if you keep digging down you will find a large cave so if you strike a large body of water, you should just keep digging down this has always worked for me and if this doesn't work you can try to branch of and drain the water through an extra cavern. This worked for me and might work for you but I am not completely sure.

Feedback appreciated!