Unless you like aggrevating people, enjoy fighting against the wind and also want to appear a bit douchy, nothing. The ones, who download - they probably like your music. Or why would they do it? Take solace in that. They might or might not want to drop money for it, but one thing is clear - they aren't going to if you'll go all law and order on them. And if you're in it solely for the money - you deserve none of it. Instead, set up bandcamp, soundcloud, some kind of charity, perhaps. Engage with your listeners, promote your stuff around, use money for advertising and awareness rather than ligitations and lining shyster's pockets. From time to time remind people that you're not really using the money to wipe backside (well, unless you're rapping - then things along these lines sort of belong to the genre). And so you'd appreciate that they cough up for things they like, that every little helps and such; but don't press on it, because, likely, neither do they bathe in benjamins, you know.
Oh, and the thing about dealing with that website - as long as there's single digital copy out there somewhere, it's going to pop up somewhere. Depending on the nature of that site (if it's not filesharing malware ridden cesspool of pron, but something you can see yourself associating with), perhaps you can make a deal with that site. Take note how that works with some TPB indies. I've seen RuTr....rrrr also doing this and it's nearly always a great success.
On the other hand, if you aren't convinced - take note how well it has worked out for the big labels. Nobody really gives a crap about their posturing, warnings and other bull. People just pump, dump, rip, share around and listen to the stuff anyway. If they're gonna, they're gonna.