...IMHO, not for the reason most people think. I favor standard NICS BGCs for FFL sales and voluntary but free or nominally priced BGCs done by law enforcement as a service to the transacting parties in private sales. I also support voluntary registration being available at a local/county law enforcement office. The campaign to end drunk driving had great success by trusting the individual driver to do the right thing. A lawful owner who is selling a gun today to a stranger would likely want to know he's not selling to criminal. A criminal in possession, unlawful owner or person who simply doesn't care won't be deterred with another law.
There is a lot to be said for a government that trusts the people and not much for the opposite.
With 300,000,000 guns in the US, an unmanageable number are already in criminal hands. Law enforcement is working on recovering those.
I figure if you give the average Joe a way to help, he will. Most people don't have evil as a end goal and being empowered to work against it, they will. Of course if you're a pro-control extremist and you view RKBA folks as opposition, it's not a far stretch to think of the RKBA group as "the enemy". And since "the enemy" is generally equated with evil this may be a tough sell to the control side.