In Hamlet, how responsible would you say Gertrude is for her own fate?Because I think she's a very shallow creature with very little understanding of the world around her.
She is blind, like Jocasta in Oedipus. She is blind to her husband's murder, her new husband's treachery, her son's melancholy, and the personal and political upheaval in her state of Denmark. She lives in complete denial as a coping mechanism, I suppose. She is so used to being Queen that, after her husband is murdered, she wants to quickly resume the role again. So, she marries Claudius.
She brings ruin upon the marriage by committing adultery and incest by sleeping with Claudius while King Hamlet was alive. This also invites Claudius to murder his brother, so that he may steal his crown and wife. Gertrude baits herself and wants to be coveted.
She recognizes, after Hamlet angrily confronts her, the source of his problems. She says I think he is sad for his father's death and "our o'er hasty marriage." (Really? It took you until Act II to figure this one out, Mrs. Obvious?) She willingly lets Hamlet be sent to England, thinking this will solve his melancholy, not knowing that he was being executed. Much of Hamlet's rage toward Ophelia in his "get thee to a nunnery" harangue is directed at his mother. And I think she deserved, and I wish she had been there to hear it.
She does not realize she is sleeping with the enemy until, of course, she is poisoned by him in Act V. Her fate, by then, is sealed. She gets caught in the crossfire, but she is no innocent victim. By not being a better wife while her husband was alive, and by not being a more responsive mother to her grieving son, and by being blinded by denial, she seals her own fate in this tragedy.