Republicans because it is pro-law-enforcement, anti-crime.
Democrats because it is a further centralization of the government bureaucracy.
If DNA testing was left up to individual cities and/or states then those communities that were willing to pony up the funds would tend to have lower crime rates than those that didn't.
And if the local communities were the ones that were funding it they could limit the collection of DNA samples to just those who were convicted of crimes.
But even Republicans are against taxes at any level of government, so there will never be enough funding at the state or local level, and our bureaucracy will continue to be centralized in DC.