In the matter of proving the existence of God, with whom does the responsibility lie with- the person who claims there is a God or the person who claims there isn’t? Also, if a person was sincerely and openly asking you for proof of God, what would/could you offer them?

The burden of proof in a particular discussion may depend on the format. If someone claims that God exists, another person may ask for some demonstration of the fact. Actually bhakti-yoga, Krishna consciousness, does demonstrate Krishna, and that’s why we all joined the Hare Krishna movement.

The claim that there is no God is virtually impossible to demonstrate. You may want to listen to my talk on atheism which you can find here. A person sincerely asking for proof of God would have to show that sincerity by undertaking the proving process which is the practice of Krishna consciousness. Otherwise, the person is not a serious investigator of God.