Given the traditional view of God and God's role in the creation of the universe, it is clear that God is at least partly responsible for evil. Without God, there would be no evil, but God may not be the sole cause. If humans have free will, then humans have a share of the responsibility as well. The question ought to be rephrased: "To what extent is God responsible for evil?"
An analogous situation can be found if one asks who is responsible for a car accident. To begin with, had the car never been manufactured, then the accident could not have occurred. At least to this extent, the creator of the car is responsible. Further, there may have been a defect in the construction of the car, which would give even more responsibility to the maker. However, it is possible that the driver of the car could have avoided the accident, and therefore the driver shares in the responsibility.
Returning to the original question, we see that God, as creator, is responsible for evil just by creating humans that do evil. As well, if the human tendency to be evil can be likened to a defect in humans, then surely God is responsible for this defect, and therefore responsible also for the resulting evil actions. The rest of the responsibility belongs to the individual who does evil, provided they have free will.
Since God created all things, and gave humans their bodies, their world, their inclinations and their desires, it is likely that God is largely responsible for evil. Humans cannot be wholly absolved of the responsibility, but even given free will it seems that humans are playing in a game with the deck stacked against them -- we seem designed to seek out evil, and only God can be blamed for that.