This is about spiritual death, also called the second death in Rev 2:11 (although physical death was also a consequence of original sin). Death is a consequence of sin (see Romans 6:16). Thus, the devil had power over humanity through the power sin had over our lives. As the letter to the Romans said, we were once slaves to sin— and that means death. The passage in Hebrews says that Jesus was born into humanity and died so that he could take back the power that satan had over us. He freed us from the slavery to sin and it’s consequence of eternal death. So, satan no longer has the power over us as long as we accept Jesus’ free gift of salvation.
Regarding physical death, well, that is just a consequence of original sin we all still have to deal with. You could say that the devil had a role in our physical deaths by way of original sin. But he does not generally “cause” us to die… that is just part of human nature now. We have finite bodies. In the beginning, I suspect that Adam and Eve had glorified bodies (or something like it) and they gave that up for all of humanity when they sinned. At any rate, Jesus did not “fix” that for us and so it is appointed to all mankind to endure physical death. If Jesus had intended to remove the consequence of physical death then I suppose he could have— and baptized believers would not die. However, that is not God’s plan. As far as God’s power over death and satan’s power St. Thomas Aquinas said, “A judge has one kind of power of death: he can punish people with death; a criminal has a different kind of power of death–a power he usurps by killing another …]. God has the first kind of dominion over death; the devil has the second kind, for he seduces man to sin and leads him to death.” This was quoted in the Navarre bible study for Hebrews.