That's an excellent question. The answer depends on exactly what you mean by "better than chance." If you mean "more than half of those convicted of a crime are guilty of that crime," then I'd say yes, there is excellent reason to think that they are. Prosecutors usually have access to several times more reports of crime than they can afford to go out and prosecute. Prosecutors are often explicitly or implicitly evaluated on their win ratio -- they have strong incentives to pick the 'easy' cases where there is abundant evidence that the suspect is guilty. Most defense lawyers will cheerfully concede that the vast majority of their clients are guilty -- either the clients admit as much to their lawyers, or the clients insist on implausible stories that don't pass muster, which the lawyers have to disguise in order to get their clients to go free. Although as a matter of law and rhetoric people are presumed innocent until proven guilty, as a matter of cold statistics, someone who has been lawfully indicted in America is probably more likely to be guilty than innocent. In fact, there are probably so many guilty suspects in Court that the legal system does strictly worse than what social scientists call a "naive predictor" -- i.e., just assuming that everyone is guilty. Of course, that wouldn't be a sustainable policy -- prosecutors choose easy cases because they know that they'll be required to win those cases in a relatively challenging environment. If the rule were that everyone is guilty, prosecutors would start choosing cases based on other criteria, and the percentage of indicted suspects who were actually guilty would go down.

Suppose you survey defense attorneys, and conclude that, say, roughly 80% of indicted suspects are guilty. Could you somehow measure whether the legal system does better than a "mixed strategy predictor" that guessed that a suspect was guilty with probability 0.8 and guessed that a suspect was innocent with probability 0.2? The mixed-strategy predictor would get an accurate result in (0.8) ^ 2 + (0.2) ^ 2 = 68% of the time. To assess whether the legal system is better than a mixed-strategy predictor, you would need to have a way of validating at least a sample of actual cases. I really have no idea how you would start to do that. It's not clear that self-reported guilt or defense-attorney-assessed guilt will correlate strongly enough with actual guilt that we can figure out which individual cases the legal system gets right and which ones it gets wrong. But if you can't measure accuracy in individual cases, how do you figure out the system's overall accuracy rate? It's not clear that looking at appellate results or DNA exonerations, etc. would help either. A reversal on appeal is no guarantee of innocence, because a sentence can be reversed (a) if the evidence is still strong but not strong enough to remove all reasonable doubt as well as (b) when the prosecution or police have used inappropriate but reliable tactics (such as using high-tech cameras to take pictures of the inside of your home without a warrant).

Finally, there is "better than chance" in the sense of specific forensic techniques being verifiably better than, say, a Ouija board. There are several pretty good techniques, such as document analysis, DNA analysis, electronic tracing, and perhaps even paired-question polygraph testing. Whether or not the system interprets the evidence correctly, a typical trial at least contains sufficient evidence for a rational evaluator to beat chance.