This is where it's important to frame the argument in terms people who worry about restricting the rights of the mentally ill can relate to.
This issue came up on my facebook feed - my wife has a lot of very liberal friends - in the form of an article claiming that mental illness was not the cause of mass shootings/
My response was to point out the actual percentage of mass shooters with a documented mental illness (25%) and the percentage of mass shooters that had psychological "stressors" prior to the shooting (62%, based on FBI data. I then also make the obvious suggestion that any one willing to commit a mass murder probably qualifies for a DSM-V diagnosis even if they've never formally received one - it is not normal human behavior.
I did agree with the article however, that liberals on the feed that mental illness itself was not a predictor of violence - pointing out the millions of people with some form of mental illness diagnosis who are not violent and do not commit crimes.
However I also pointed out that there are about 100,000,000 gun owners in the US, and about 300,000,000 million guns. I then did the math to show the percentage of gun owners and guns that were not involved in any kind of gun related homicide. There are roughly 15,500 murders or non negligent homicides per year in the US, and firearms are used in about 72% of them. That's 11,160 gun related murders. Even if each one was committed by a unique individual that means only 0.015% of gun owners will commit murder. And it means that gun control efforts are restricting the other 99.9845% of gun owners who will never commit a murder.
Then I compared the 25% of mass shooters have a DSM-diagnosis before the shoot, and 62% have a significant psychological stressor. I then asked whether it's fair to extend protection to people with mental illness to ensure they are not denied their rights, and not do the same for the 99.9845% of gun owners who also have posed no threat to anyone, when their rates of mass shootings are orders of magnitude lower.
I got agreement on that - from liberals.