I don't think it would doom the internet, but rather give mods an incentive to not allow illegal things to happen on their websites
It would give any sensible person reason enough to not participate in the moderation of any forum or website that may come under scrutiny by any relevant authority. Fear is not in and of itself incentive to do or not do anything. Reward is much more of an incentive.
I don't want to be the mod of any forum on bitcointalk.com (not that anyone has asked) for fear of being charged with facilitating money laundering. Reasonable compensation may incentivize me to perform the duty as expected.
Say public loitering and skateboarding is illegal in a particular jurisdiction. Should the shop owner forsake providing parking space for "legal" patrons for fear of being charged with aiding and abetting the wrongdoers?
I haven't looking into the DPR/Silk Road case in any great length. I do not think the owners of craigslist.org, bitcointalk.org, ebay.com, amazon.com, reddit.com, facebook.com, Silk Road.onion, etc. should be held responsible for users' actions. It just doesn't make any sense. People commit crimes. Websites are just there for people to use and abuse.
We have enough moderation and censorship of the internet in the "free" world. Laws, as they are written, have long arms and great scope. We should be actively looking to reduce this reach rather than increase it. The children are fine. The terrorists don't really hate most of us. And, the elderly are oblivious. Let's just be allowed to live and let live.