Iago, of course, is primarily responsible. He calls attention to Cassio and Desdemona talking together innocently and suggests that something more is going on between them. He tells Othello that even though he does not want to speak his thoughts, he does think that Othello might watch his wife with Cassio a bit more closely. He suggests that Othello does not really know how Venetian women act, and reminds Othello that Desdemona deceived her father and is therefore capable of deception. He then clinches his argument with the "ocular proof" of Desdemona's handkerchief, telling Othello that he saw Cassio wiping his beard with it. For the next proof, he mentions that he overheard Cassio talking in his sleep, saying "Sweet Desdemona, let us hide our love." So, Iago is responsible for planting the seeds and manipulating Othello to think Desdemona is guilty of an affair.
Emilia is only slightly, if any, culpable. Emilia sees that Desdemona's handkerchief has fallen to the floor when Desdemona and Othello were talking. Emilia picks it up and gives it to Iago, because Iago had been pressuring her to steal it. To Emilia, the handkerchief means nothing. She doesn't even want to give it to Iago until he tells her what he is going to do with it. But he snatches it away from her and plants it in Cassio's lodging, so that it will seem as if Desdemona put it there.
Later, when Desdemona is wondering where her handkerchief is, Emilia remains silent. Emilia does not know what her husband has done. She has no idea that Iago is using the handkerchief to frame Desdemona and Cassio.
Their relationship deteriorates as Iago drives the idea of an affair between Cassio and Desdemona deeper and deeper into Othello's brain. The other factor that many folks have pointed to is that Othello suddenly has no purpose and no confidence because there is no war to fight, the only arena in which he has ever felt at home and confident. So as he tries to deal with being at peace and the intricacies of the peaceful life, he really struggles and it helps to cause friction.
In terms of responsibility, Iago clearly is trying very hard to create the problem, and I suppose you could hold Emilia responsible except that the actions she takes (handing over the handkerchief) she takes unknowingly so I would hesitate to say she is the one responsible.