ducklings can swim after a few days old, but they must only be in the water for a short time, as their down absorbs the water, and they get chilled very quickly. They must be supervised at all times, and the water shallow.
I let mine swim when they're a couple of days old in water deep enough they don't touch the bottom, but I'm right there with them and I limit their swim time so they don't get waterlogged and chilled. Make sure they get under the heat lamp to dry out well after their swim.
When I asked this and Katy said she lets hers swim at a couple of days, I decided to try it. Mine are only a few days old and the past two days have had supervised swim (and poop LOL) time in my bathtub. They LOVE it and it is soooooooooo cute!!!! I have the water shallow enough that they can stand at any time and I don't even turn my head for a second. I treat it just like when I was bathing my children as babies.