My understanding is that Satan and his minions still have influence in this world, but they have been significantly incapacitated with respect to their power over our souls. In the Thanksgiving Prayer of the Coptic Hourly Prayers, we thank Christ who has "given us the authority to trample on serpents and scorpions, and over all the power of the enemy." Apa Shenoute of Atripe depicts the achievement of Christ and hence the situation at present, in rather vivid imagery:
For like a righteous king who has cast down his enemy in battle and made him weak, and who has said that all those who are on his side should lay their hands upon him [the enemy] because he wants them all to be glorified together, and who later will return and cut off the enemy's head--just so, and even more, when the Lord Christ came, he destroyed the devil like a tyrant whose legs were cut off up to his thighs and his arms up to his shoulders; and as for the other members of his body, his heart and his spine, he [Christ] struck them all. He [the devil] is unable to move [any of his members] so as to get up or pursue a person, except for his breath alone, which comes and goes, that is, his thoughts, which Christ left in him because he wants his children, his soldiers, his servants, and all who are on his side to lay their hands upon him [the devil], that is, to fight against his goldess thoughts [logismoi], so that they might be glorified with him and reign with him.