You don't want to use straw in a brooder, it'll stink to high heaven. Pine shavings would be better, more absorbent.
If you can cut a hole in the bottom of the brooder, say 12x12, and cover it with hardware cloth, to place the waterer on, that will solve the water issue. Place a bucket below to catch the waste water. It will keep the smell down a lot and will keep everything dry.
You could use straw if the bottom is all wire, it can air out from the bottom and water can drip through. I wouldn't use straw on a solid floor with ducks unless it had the shavings mixed in too. Course I'm not a fan of mucking the whole thing out everyday.
Just depends on your brooder set-up really. Finding a way to contain the waste water will be the biggest help. I've used everything from placing the waterer into a paint roller tray, setting it into a larger pan, but so far, having it sit on wire has done the most good, because they can't replay in the waste water and throw it all over.
It's the water + wet food + poop + heat from the lamp that causes the stink. Not so bad if you find a way to manage the water.