Paul states that the law has "dominion" over a man only as long as he lives. Some have interpreted this to mean that, now that we have died with Christ, the law is no longer binding on Christians. Indeed, some modern translations of the Bible translate this verse to say just that. However, note how Paul uses this word "dominion" in other places.
In Romans 6:9, Paul speaks of Christ's immortality now that He has been resurrected, saying, "Death no longer has dominion over Him." During the period that Christ was a flesh-and-blood human being, He could die, and He did die on the cross. Now, however, death no longer has any power over Him, because He is an immortal Spirit Being.
In Romans 6:14, Paul uses the same word to describe our relationship with sin. "For sin shall not have dominion over you." Here he shows how our past sins have been forgiven, and we have access to Christ's atoning grace for forgiveness of future sins. Therefore, sin no longer has the power to condemn us to death.
Throughout Romans 6 and 7, the Greek word translated "dominion" is kurieuo, meaning "exercise lordship over." Paul uses this term in the context of having power over something. In Romans 6:9 and 14, he states that death and sin no longer have power to harm us or to cause any adverse effect in our lives.
Now we can better understand Paul's meaning in Romans 7:1. In this verse, Paul explains how the law has "power" over a human being only while he lives. He means the law has power to condemn us as a sinner and, consequently, condemn us to death only as long as we are alive. Once we have died, the penalty for sin has been paid, and the law has no more power to condemn us.