Right now, provided a serial number, American police can trace a gun used in a crime to its last legal owner. It takes a number of phone calls -- first to the manufacturer, then to the gun store, then to the first owner, and down the chain -- but it generally works.
And, although it's not mandated by law, gun owners do tend to keep records on private party sales. Nobody wants to be in the situation of "we found this Glock at a crime scene, and it appears to be yours?" and not be able to provide evidence to the contrary.
So a registry doesn't really offer any real utility... except to confiscate those weapons down the road.
If you want to promote safe storage, the best solution would be a tax benefit for buying safe-storage equipment. Gun safes are expensive, at least if you want something that can't be crowbarred open in a few minutes.
Something else that I would love to see, and stealing a page from the Swiss, is universal gun education.
There are more guns than cars in the United States, and I find it negligent that not every American learn not just gun safety, but also how to safely store, handle, and yes, fire a gun -- even if they never plan on owning one.