Background checks involve checking that a prospective gun owners fit a certain set of criteria, such as not having a criminal record or history of mental illness. A gun is usually safe in the hands of a 50-year-old woman. It is not always as safe in the hands of a 20-year-old convicted felon. Therefore, we should not limit ownership to people with a record of responsible behaviour. But those with criminal records or records of mental illness should not be permitted to own a gun.
Controlling who has access to firearms is essential for reducing gun-related deaths. But in some states, there are more checks on those who want to adopt a dog than there are on those who want to purchase a firearm. Restricting gun ownership in cases where the owner has a criminal history or a history of mental illness works. The best example of this is in the cases of Connecticut and Missouri. Connecticut tightened licensing requirements for gun ownership in 1995. Missouri, on the other hand, replaced many licensing requirements in 2007, making it easier for residents to purchase firearms. The result was clear. In Connecticut, gun homicide rates fell by 40%, while in Missouri, they rose by 25%. Gun suicides also decreased by 15% in Connecticut after the introduction of more stringent background checks. In Missouri, they climbed by 16%. Background checks are also extremely popular with the American people: over 90% support background checks on all gun sales.
You cannot pick and choose who is entitled to receive constitutional rights. If you accept the Second Amendment is a right granted to all Americans by the constitution, then you can't tell some people they can't have that right just because they made a mistake when they were younger or suffered from a mental illness at one stage of their life. Also, there is no indication to suggest that background checks would prevent mass shootings. Nicholas Cruz, the Parkland shooter, passed a background check to get his AR15. In fact, 80% of weapons used in mass shootings are obtained legally through the background check system. 
Individual rights are not unassailable. Only certain people can be trusted with certain rights.
[P1] Gun homicides are reduced when background checks prohibit felons and those with a history of mental illness from owning guns. [P2] Therefore, the government should carry out background checks to prevent these demographics from owning firearms.
Rejecting the premises
[Rejecting P2] If you accept the Second Amendment as a constitutional right, then you have to accept that it applies to all Americans. You cannot pick and choose who it applies to.
Moore, Mark H. “The Bird in Hand: A Feasible Strategy for Gun Control.” Journal of Policy Analysis and Management, vol. 2, no. 2, 1983, pp. 185–195. JSTOR, www.jstor.org/stable/3323282.