Frankly, according to the Bible Christians are NOT stronger than Satan, and that is not because he is a spiritual being and they are physical/spiritual beings. If anything, having bodies makes Christians more vulnerable to his attacks, not less. As the apostle Paul observed, our bodies have "members," and since our members war against each other (Romans 7:23) Christians must allow "the law of the Spirit of life in Christ Jesus to set . . . [them] free from the law of sin and death" (Romans 8:2).
There is nothing inherently wrong with the human body, since it has been designed by God, but physical senses and appetites of necessity and by God's design interact with bodily members, sometimes in God-pleasing ways and sometimes not.
First John 4:4, quoted already in an answer, is a good starting place, but the verse needs to be unpacked a bit!
Notice in this verse that the Spirit (capital S) is greater than the spirit (lower-case S) who lives in the world (viz., the devil). The Holy Spirit, who indwells every believer, is the Spirit of Christ. Through Christ's Spirit within them, Christians are "more than conquerors" in spiritual warfare (Romans 8:37), since he fights on their behalf. Their duty is simply to stand in the power of his might (Ephesians 6:10). Jesus said,
"Without me you can do nothing" (John 15:5)
To grow and thrive in Christ, Christians must "abide in Christ" as he abides in them (John 15:4 ff.). Abiding in Christ requires Christians' heart commitment; it takes spiritual awareness of the warfare going on around them and within them; and it takes unbroken fellowship between them and Jesus, which requires frequent confession and repentance.
Yes, believers can choose to surrender control to the devil and his influence, but without an awareness of their vulnerability to his attacks they can surrender more control to him than they think they do. Forewarned is to be forearmed. Knowing that the devil prowls about as a lion seeking to devour true believers (1 Peter 5:8) requires spiritual diligence on their part. Do they therefore need to fear the devil? Yes and no.
Yes, Christians need to fear the devil because he is powerful, hateful, and well organized. His "world system" exerts a powerful influence on earth, and believers need only to read the headlines of today's newspapers to realize he is indeed very active in wreaking havoc in people's lives and in society at large.
Jesus likened the devil to a sheep-stealer who comes only to "steal and kill and destroy" (John 10:10). Jesus, on the other hand, comes only to give his children an abundant life. That abundant life is within reach of every Christian, but sometimes spiritual strongholds and besetting, entangling sins in their lives (Hebrews 12:1) short circuit that abundant life in Christ.
For that reason, believers must come back again and again to the cross of Christ, where Satan was defeated. As the old hymn puts it,
Nothing in my hand I bring,
Simply to thy cross I cling . . ..
So no, Christians need not fear the devil. When they cling to the cross of Christ, crucifying their old natures by taking up the cross daily (Luke 9:23), victorious living for them becomes a reality as the atoning blood of Christ and their whole-hearted witness to the truth of God come together through faith. Revelation 12 says the following:
Then I heard a loud voice in heaven, saying, "Now the salvation, and the power, and the kingdom of our God and the authority of His Christ have come, for the accuser of our brethren [viz., Satan] has been thrown down, he who accuses them before our God day and night. And they [i.e., believers] overcame him because of the blood of the Lamb and because of the word of their testimony, and they did not love their life even when faced with death" (vv.10-11 NASB Updated, my emphasis).
The blood of the Lamb is the atonement to which you referred in your question. The word of their testimony comprises the truths to which Christians cling, truths which assume 1) that their testimony is informed by the truth of God, and 2) that they are prepared and willing even to die for that truth.
In conclusion, Satan is a formidable foe. To fail to recognize this truth is to underestimate his influence and power, and failure in this regard opens believers up to his influence and attacks. The solution is to put on by faith the complete armor of God (Ephesians 6:13-17):
God's truth, which guards the loins
the breastplate of righteousness
battle shoes of the preparation of the gospel of peace
the shield of faith
the helmet of salvation
the sword of the Spirit
Having done these things, believers are then simply to "stand firm" (v.14), which means they are not in attack mode; rather, they are dependent on God and his armor to defend them.