Attorneys General of 47 states sent a letter to Congress in July of 2013 recommending that the civil and criminal immunity in Section 230 be removed. So there is broad support for doing something to address internet companies' responsibilities, but it is hard to find agreement on what should be done.
The ACLU came out in opposition to weakening the law's protections, and went so far as to submit a rebuttal letter to Congress one week later.
If their proposal were to pass, it would mean that every website on the Internet could be subject to legal liability for violations of an unfathomable number of state laws.
Matt Zimmerman from the Electronic Frontier Foundation had this to say about The AGs proposal:
Their approach is wrong, and dangerously so, but even if the AGs disagree and want a debate about how state criminal laws fit into the regulation of the Internet, they owe the public a more honest discussion.
Instead, the AGs are really proposing to do something far more revolutionary in scope: make service providers criminally responsible for what their users do, even if they don't intend for any illegal activity to take place on or through their services (or even have specific knowledge about it).
It's not clear if the ACLU or EFF are against modifying the law in any way, or if they were just against the AGs specific recommended course of action.
Congress did vote in March of 2018 to narrowly limit the scope of Section 230, but the limitation was mostly concerned with service providers who 'knowingly facilitate' sex trafficking with the apparent target of the restriction being the online service Backpage.com. The bill was signed into law by President Trump on April 11th, 2018. From the debate prior to the vote:
Sen. Blumenthal: This bill would clarify section 230 of the Communications Decency Act, which was never intended to give websites a free pass to aid and abet sex trafficking. It was never intended to immunize completely those websites so they could knowingly facilitate sex trafficking. Those words are in the bill--``knowingly facilitate.''
The purpose of our measure, very simply, is to give survivors their day in court. Right now, the courtroom doors are barred to them, as a recent court of appeals opinion remarked, outrageously so. It would also open avenues of prosecution to law enforcement where they are currently roadblocked.
It should be noted that Section 230 is not ironclad. Some internet publishers have been found liable in court for "encourag[ing] the development of what is offensive."