The answer to this question is a matter of opinion, although that is not to say there is much to recommend in either Rasheed or Assef. Both are misogynists and bullies.
In A Thousand Splendid Suns, Rasheed ignores the pleadings of his wife, Mariam, when she begs him not to disgrace her by taking another wife. However, the lustful Rasheed proclaims that he should be given a medal for marrying the fourteen-year-old Laila; after all, Mariam would finally get some household help and they would be able to take pride in saving a defenseless teenager from the streets of Kabul and the brothels of Peshawar. With this sly rationale, Rasheed forces Mariam to acquiesce to his wishes.
As a character, Rasheed has no redeeming qualities; he is abusive to both his wives, he regards daughters contemptuously because they are not sons, and he regards lying about Tariq's death a beneficial maneuver because it secures him Laila's hand in marriage. After both Laila and Mariam's failed attempt to run away to Peshawar, Rasheed brutally beats Mariam and Laila; even Laila's baby, Aziza, is not spared her father's violent abuse. Rasheed locks Mariam in the toolshed and Laila in her room for days without benefit of food or water. This also means that the baby, Aziza, is deprived of nourishment during the forced incarceration. Rasheed is a character who is not above abusing babies and women in order to reinforce his position of authority in his household. In the end, Mariam has to kill Rasheed in self-defense in order to prevent him from killing Laila.
In The Kite Runner, Assef distinguishes himself as a real villain from the beginning of the novel. His rape of the innocent and defenseless Hassan is an atrocious act of violence; in the story, Assef's perverted tendencies follow him to adulthood. As a Taliban commander, he continues to rape young boys; his victims include Sohrab, Hassan's son. This execrable character is also proud of having played a part in decimating a large percentage of the population of Hazaras. In the novel, he boasts about his murderous exploits in the Hazara massacre at Mazar-i-Sharif in 1998, as Amir tries to bargain with him on Sohrab's behalf.
"You don't know the meaning of the word 'liberating' until you've done that, stood in a roomful of targets, let the bullets fly, free of guilt and remorse, knowing you are virtuous, good, and decent. Knowing you're doing God's work. It's breathtaking."
Assef describes his participation in the pogroms in religious language; to him, it is a divine calling to assassinate those he considers inferior Muslims. He even relates gleefully that he enjoyed leaving the bodies of dead Hazaras to the dogs. Also, as a Taliban commander, Assef regularly participates in public stonings. When Amir bargains for Sohrab, Assef brutally attacks him with his stainless steel brass-knuckles. It is Sohrab's slingshot which eventually saves Amir from certain death at Assef's hands.
So, how do we decide which character is the greater villain? If you think that participating in massacres against a religious minority (plus exploiting and sexually abusing children) constitute the very height of evil, my best advice would be to pick Assef as the more villainous of the two. If, however, you think that physically abusing women and infants constitutes a greater crime, pick Rasheed. Either way, as long as you provide evidence for your assertions, you will be well on your way to answering this question adequately!