Lets start with enforcing existing laws and start arresting criminals who get flagged at gun shops while trying to buy a gun. That's one heck of a start.
This is one of the issues that the NRA has been touting for years. The NRA wants this enforced. The problem with this is what Ben Towe pointed out.
For example, shortly after the Lautenburg amendment was passed there was a cop in NY, 20 year veteran, who was denied purchase. He had been in a fight with his dad at 19, and was charged with assault. It was listed as domestic violence, but he had no idea. Imagine if they added injury to insult and slapped him with a felony for trying to buy the gun.
The bigger problem with this is that it will discourage gun ownership altogether. I know several people who have had issues in the past, and aren't sure if they would pass the background check. The threat of a felony arrest would scare them away from gun shops altogether.