I think to state she is "responsible" is to go too far. Desdemona is not at fault in having caused her own murder. It might be better to suggest that her innocent actions, based upon the friendship she harbored for Cassio, were used by Iago to manipulate Othello, whom Iago knew would misinterpret them if given the correct guidance. It is true that if Desdemona had not been fond of Cassio, she probably would not have prevailed upon her husband to be once again "as friendly as you were" with Cassio. But then, possibly she might still have been willing to put Cassio's suit forward—after all, she knows him to be a "valiant" man, and she is even urged on by Emilia to try to influence Othello, as Emilia believes, ironically, that her own husband, Iago, is troubled by what has happened.
There is no evidence to suggest that Desdemona has any romantic interest in Cassio—Iago tells Roderigo that she is "directly in love with" him only to serve his own ends, not because he really believes it to be true. Moreover, there is no evidence to suggest that Othello had ever had cause to question Desdemona's relationship with Cassio before Iago deliberately took it upon himself to have her misrepresent that relationship to him. Iago causes Cassio to lose his place; Desdemona appeals for him, based on Cassio's request—at Iago's suggestion. Iago determines that he is going to tell Othello that Desdemona is "false" with Cassio; Iago interprets for Othello Desdemona's innocent acts as false ones. Iago turns Desdemona's well-meaning gestures into things she did not intend, and Othello ultimately murders her. None of this would have happened based on any of Desdemona's actions alone. She is, more than many of Shakespeare's tragic heroines, absolutely the victim in this case.