In Mary Shelley's Frankenstein, what sense of responsibility does Victor begin to feel toward the monster ?
Victor first feels responsible in terms of the murders of William and Justine. There is a theme here of science vs nature or God. Had Victor not created the creature in the first place, those two lives would not have been lost. During Justine's trial and subsequent execution, he is haunted by self-loathing and a sense of responsibility for these two tragedies. Had he not "played God," this would not have happened.
Another of the themes in the novel is parental irresponsibility. Victor created a life and then abandoned it. Though Victor is horrified by the appearance of the creature, and feels guilty for his foolish hubris which allowed him to think he could create life, the creature appeals to him with regard to his loneliness.
A third theme in the story is rejection and abandonment. The creature speaks to Victor not only of Victor's rejection, but that of society as a whole. The creature is lonely: there is no one on earth like him, and he needs the companionship of others, just like any man. He begs Victor to make him a mate, and out of a sense of compassion and responsibility (after all, the creature never asked to be created), Victor agrees. He even begins the process. However, soon he becomes disgusted by what he his doing, and fears that the monster and his mate could wreak havoc on the world, and so he stops and destroys the progress he has made.
The creature becomes enraged and promises to destroy all those that Victor loves. In fact, it is after the murder of Elizabeth that Victor's sense of responsibility changes. It is at this point that he promises to track the monster down and destroy him. (This is, of course, what the creature wants. Hatred from Victor is better than no attention from anyone. For the creature believes he was not created to hate and kill, but has been driven to it.)
In these ways, Victor has a sense of responsibility for the monster he created.