Many women who object to submission focus on the leadership of the husband and overlook the responsibility it carries with it.
The Responsibility of Leadership
Note this sequence of events:
Genesis 3:6: "When the woman saw that the fruit of the tree was good for food and pleasing to the eye, and also desirable for gaining wisdom, she took some and ate it. She also gave some to her husband, who was with her, and he ate it."
Adam was standing next to Eve when she ate the fruit, but he didn't stop her. He joined her. I encourage you to read all of Genesis 3 to fully understand the context.
1 Corinthians 15:21-22: "For since death came through a man, the resurrection of the dead comes also through a man. For as in Adam all die, so in Christ all will be made alive."
Romans 5:14: "Nevertheless, death reigned from the time of Adam to the time of Moses, even over those who did not sin by breaking a command, as did Adam, who is a pattern of the one to come."
Even though Eve was the first to sin and she tempted Adam to sin, Adam is always designated as the one through whom sin and death entered the world.
He tried to blame Eve, but God held Adam responsible because he was the leader of his family and he failed to lead. For more perspective on Adam's blame game, see The Fourth Oldest Sin: It's Your Fault.
Don't misunderstand: Eve was fully responsible for her personal sin, but Adam carries the responsibility for the fall of mankind because he failed to lead.
A good explanation from Answers in Genesis:
the leader it was Adam’s responsibility to protect and provide for his
family. This leadership role was clearly demonstrated when God talked
with Adam and Eve following their sin. Even though Eve sinned before
Adam, God questioned Adam first (Genesis 3:9). This was because of the
leadership role God ordained for husbands in marriage. Adam, as the
leader of the family, was held responsible, not only for his own sin of
eating from the tree, but also because he did not provide adequate
protection for his wife, allowing her to sin. In the New Testament, Adam
is clearly presented as both progenitor and representative of all
mankind…Just as sin and death came through one man, Adam, it is also
through one man, Jesus Christ, that grace and righteousness are given as
free gifts to sinners. The naming of Christ as the Last Adam and the
multiple comparisons of Adam to Christ would be invalid and nonsensical
if original sin came through Eve…Interestingly, although Eve was the
first to sin, the solution to sin came through “her Seed” (Genesis 3:15). The Seed, Jesus Christ, was born of a virgin named Mary. He paid
the price for sin and will redeem those who receive the salvation He
offers…Scripture makes it clear that that sin and death entered this
world through Adam, and thus he is to blame for original sin.”
A Military Analogy
My husband was in the military for 22 years. When he was a company commander, each soldier in his company was held responsible for violating orders or breaking the law. However, if a soldier's actions affected a mission of the company, my husband was also held responsible.
There's an acronym used in the military: RHIP, Rank has its privilege. But there's an acronym not used that is just as true: RHIR. Rank (leadership) has its responsibilities.
My husband and I recently made some important decisions about our future. We discussed these issues as equals. We debated different aspects, we prayed, and sought counsel. But in the end, my husband made the final decision, and he will bear the responsibility for that decision in a way I will not.
I consider that a privilege and a protection husbands don't have. Adam was held responsible for the sin of mankind in a way Eve was not. And our husbands are held responsible for our families in a way we are not.
In addition, women are given an incredible position of influence with our children and husband, and that influence is every bit as important as the husband's leadership. Eve used it for evil, but we can use it for good. To read more, see For the Love of a Woman.