Does the Bible teach that Christians can bind Satan and demons?
An examination of how the Bible connects the terms "binding" and "loosing" to the apostle's unique role in founding the church will help you determine if believers can or should bind Satan.
Jesus gave the apostles sole authority to bind and loose things on earth. In Matthew 16:18-19 Jesus says, "You are Peter, and upon this rock I will build My church; and the gates of Hades shall not overpower it. I will give you the keys of the kingdom of heaven; and whatever you shall bind on earth shall be bound in heaven, and whatever you shall loose on earth shall be loosed in heaven."
Later He said essentially the same thing to the rest of the apostles (18:18). Ephesians 2:20 tells us the church is "built upon the foundation of the apostles and prophets, Christ Jesus Himself being the corner stone."
The apostles' ministry was foundational--they constituted the authority and formation of ministry within the early church as directed by the Holy Spirit. There's an example of that in Acts 15, where the apostles and elders of the church working through perhaps the most significant issue they had dealt with thus far. Some people associated with the church claimed circumcision was a requirement for salvation. The apostles' decision against that position became binding on all the churches (Acts 15:22-31). The Holy Spirit orchestrated their decision according to God's will (v. 28).
After Jesus commanded His disciples to receive the Holy Spirit in John 20:22, He told them, "If you forgive the sins of any, their sins have been forgiven them; if you retain the sins of any, they have been retained" (v. 23). He was not giving them the power to forgive sins--only our Lord Jesus Christ can do that (Mark 2:7-10; Acts 4:12). Rather, He gave them the authority to declare what God has already done in heaven (cf. Matthew 6:10).
Jesus gave the apostles the authority to bind and loose--speak and act under God's authority--as the foundational representatives for the church. They did not act arbitrarily, nor did they operate apart from the Holy Spirit (Acts 2:42-47; 4:28-33).
Some misapply that teaching to include binding Satan. There is no scriptural command to bind Satan, nor is there any biblical example of the practice. Satan remains at large as the "prince of the power of the air" (Ephesians 2:2) until he is chained or bound (by an angel, not a human being) during the millennial reign of Christ (Revelation 20:1-3). The disciples cast out demons, but they never bound them or Satan.
For further information, consider John's book How to Meet the Enemy or his audio series Spiritual Warfare: Fighting to Win.