The frequency of these lawsuits will skyrocket. It's an easy way for lawyers to make a buck. Why? Because everything that the lawyers need to prove to support a claim in civil court is documented automatically by the torrent process.
I don't think there is a legal solution, but there is a simple technical one: Strongly encrypt the torrent payload (i.e., the movie, tv show, album, etc.) Do NOT include the password in the torrent. Instead, include in the torrent a link to a site somewhere outside US jurisdiction that has the password. That way the only thing that the lawyers could prove was that someone downloaded an encrypted movie, tv show, etc. But the lawyers could not show that the downloader ever was able to extract the copy of the movie, tv show, etc., from the torrent. The only source to show that the downloader had the password would be the logs of the site where the password was posted. If that site is outside US jurisdiction, it would be cost prohibitive for the law firm to get the evidence that the downloader obtained the password. Most likely, the lawyers could not even ask the downloader if he or she ever obtained the password, because of the Fifth Amendment prohibition against self-incrimination.
Bottom line: The addition of encrypting the torrent payload, but including a link to the password, will make these lawsuits too expensive, and they will dry up.