The Usenet server is centralized though. They could sue the provider to get the names of everyone that downloaded a set of posts. This is made much easier by the fact that most popular Usenet servers are paid for nowadays (especially ones that carry alt.binaries.*).

They would first have to pass laws to force Usenet providers to log their customers' viewing and download history.

Many Usenet providers make a 'will not track' promise to their customers. This is from Giganews' homepage, for example: "We will never sell your information to third parties or track what you download." http://www.giganews.com/

Has anyone ever been prosecuted for downloading something? Every case I have seen is about uploading, otherwise you can't even safely browse youtube--someone might have used a copyrighted song in their cat fart video.

They could. The answer they'd get is, "it's too expensive to store everything people downloaded, and our paying customers don't want us storing that information anyway."

A law could be passed requiring Usenet providers in country X to log downloads, but then you could just use a server hosted in some other country.

Preventing people from copying information is like preventing water from being wet. You fail.