If God is sovereign, is He responsible for evil?
No. Scripture says that when God finished His creation, He saw everything and declared it "very good" (Genesis 1:31). Many Scriptures affirm that God is not the author of evil: "God cannot be tempted by evil, and He Himself does not tempt anyone" (James 1:13). "God is light, and in Him there is no darkness at all" (1 John 1:5). "God is not the author of confusion" (1 Corinthians 14:33)—and if that is true, He cannot in any way be the author of evil.
Occasionally someone will quote Isaiah 45:7 (KJV) and claim it proves God made evil as a part of His creation: "I form the light, and create darkness: I make peace, and create evil: I the Lord do all these things" (emphasis added).
But the New American Standard Bible gives the sense of Isaiah 45:6-7 more clearly: "There is no one besides Me. I am the Lord, and there is no other, the One forming light and creating darkness, causing well-being and creating calamity; I am the Lord who does all these." In other words, God devises calamity as a judgment for the wicked. But in no sense is He the author of evil.
Evil originates not from God but from the fallen creature. I agree with John Calvin, who wrote,
. . . the Lord had declared that "everything that he had made . . . was exceedingly good" [Genesis 1:31]. Whence, then comes this wickedness to man, that he should fall away from his God? Lest we should think it comes from creation, God had put His stamp of approval on what had come forth from himself. By his own evil intention, then, man corrupted the pure nature he had received from the Lord; and by his fall drew all his posterity with him into destruction. Accordingly, we should contemplate the evident cause of condemnation in the corrupt nature of humanity-which is closer to us-rather than seek a hidden and utterly incomprehensible cause in God's predestination. [Institutes, 3:23:8]
It is helpful, I think, to understand that sin is not itself a thing created. Sin is neither substance, being, spirit, nor matter. So it is technically not proper to think of sin as something that was created. Sin is simply alack of moral perfection in a fallen creature. Fallen creatures themselves bear full responsibility for their sin. And all evil in the universe emanates from the sins of fallen creatures.
For example, Romans 5:12 says that death entered the world because of sin. Death, pain, disease, stress, exhaustion, calamity, and all the bad things that happen came as a result of the entrance of sin into the universe (see Genesis 3:14-24). All those evil effects of sin continue to work in the world and will be with us as long as sin is.
First Corinthians 10:13 promises us that God will not permit a greater trial than we can bear. And James 1:13 tells us that God will not tempt us with evil.
God is certainly sovereign over evil. There's a sense in which it is proper even to say that evil is part of His eternal decree. He planned for it. It did not take Him by surprise. It is not an interruption of His eternal plan. He declared the end from the beginning, and He is still working all things for His good pleasure (Isaiah 46:9-10).
But God's role with regard to evil is never as its author. He simply permits evil agents to work, then overrules evil for His own wise and holy ends. Ultimately He is able to make all things-including all the fruits of all the evil of all time-work together for a greater good (Romans 8:28).