How would a national registry prevent crime with illegally owned guns? A registry of cars doesn't prevent vehicular manslaughter/homicides at all. As
@Danoff mentioned, some 60% of crimes are done with an illegally obtained gun. Having a registry is only going to make that number go up. Nevermind that the US does a horrible job at preventing illegal guns from entering the country because we're too worried about a Mexican coming in to pick fruit.
Assuming we start a national registry that requires me to update my registration yearly (like a car), what's that actually going to do? All I see it doing is fleecing money out of me (like vehicle registration) for the privilege of owning something that's granted to me by right as an American citizen. You'd also have to ignore the horrid job the US government does with tracking anything. The government loses money, weapons, people, documents and hell, we even lost moon rocks. We also have a ton of people on visa overstays that are never addressed despite them being actual job-stealing illegal immigrants.
There's more than that, but they're not all "the owner has to do something". Some states require retailers to keep a registry of guns they sell, which essentially is a database of the owners.