Here's an argument that we shouldn't:
There are some things we shouldn't put to a vote. The majority can't vote to enslave a minority, or to confer fewer rights on some groups than others. The power to imprison is one of the more awesome powers we grant the government, and democracy is too crude an instrument to protect our rights in the face of that power.
But even here, the alternative isn't optimal. Voters are too easily manipulated by crime fearmongering and tend to reward, not punish, overly aggressive prosecutors, as well as punish judges who show the slightest hint of balance, mercy, or a better-than-narrow view of due process. But I've also written in the past about how rarely prosecutors are punished by courts, the state bar, or the state attorney general for even egregious violations, even in cases that result in wrongful conviction. Given what we already know about accountability in civil service jobs—that is, that there's very little of it—I don't know that there's any reason to think it would be much different for prosecutors. Still, making DAs civil servants would least insulate them from the need to justify their job to voters by racking up convictions, which in turn might eventually attract more people to the position whose concept of justice is a bit more nuanced than filling up the prisons with bad guys.
Another perfect example of the conflict between democracy and individual liberty. What about this idea? Should due process be subject to majoritarian control?