It would also help solve crimes to have everyone's fingerprints on file. Also, we could give have everyone give the police keys to their house so the cops can look around any time they feel like it.
The courts have found that taking a DNA sample and analyzing it for inclusion in the FBI database constitutes a search, which normally means police would need to get a warrant to do it. Persons convicted of crimes have their DNA sampled without a warrant because they lose some of their rights when convicted. Of course, the issue is being pressed ever further -- most arrestees are now DNA sampled, although there continues to be some litigation over the legality of that practice. The balance between freedom from government intrusion and effective policing is a very hard issue, especially in a modern society where sacrificing personal data for convenience (looking at You, Google) is a bargain we willingly make every day. Is it worse for some non-gene-coding DNA strings to sit in a database for years or for Google to know the time and place you searched for "Kate Upton + chipmunk costume + olive oil"?
DNA is an incredibly powerful crime fighting tool. And it's useful in more situations than murder. Under the right conditions, anything you touch can have your DNA on it -- a weapon, a doorknob, a ski mask. DNA can solve assault cases, bank robberies, burglaries--really any crime where a person was physically present. But there are troubling issues associated with it. Aside from the Fourth Amendment/search warrant issues, what does it cost to process DNA samples? State crime labs are being overwhelmed with thousands of swabs taken at crime scenes by investigators who are hoping for a lucky contact-based DNA hit. Moreover, what about other uses for DNA? If everyone were profiled, would they have the right to have, e.g., an unknown father identified? Should people be routinely notified if the genetic parents are not the people on their birth certificates? Lots of interesting issues.