First, the quote needs to be understood in light of the sociosexual relationships prevalent in the ancient world. Those relationships were hierarchical: the husband ruled and the wife was the husband's property. There was no equality between them. Paul was a man of his time; he saw the husband as ruling head of the wife, a different relationship from the kind we experience today. But he also saw with prophetic vision how a marriage can be transformed in Christ.
St. Paul writes:
"Be subordinate to one another out of reverence for Christ. Wives should be subordinate to their husbands as to the Lord. For the husband is head of the wife just as Christ is head of the Church, he himself the savior of the body. As the church is subordinate to Christ, so wives should be subordinate to their husbands in everything. Husbands, love your wives, even as Christ loved the Church and handed himself over to her to sanctify her....So also husbands should love their wives as their own bodies. He who loves his wife loves himself. For no one hates his own flesh, but rather nourishes it and cherishes it, even as Christ does the Church, because we are members of his body."
This is a quote which is often disliked because today of course we teach the truth that men and women are equal, both created in the Image and Likeness of God. And truthfully, there are some groups who use this quote to justify women being submissive and men being dominant in relationships which are abusive. But if we look at it from a different perspective we can see that very subtly it points the way to a strong MUTUAL love, an interdependence, a creative team-work..
So often we forget this first line: "Be subordinate to ONE ANOTHER out of reverence for Christ." I prefer the word "obedient" instead of "subordinate." If both husband and wife are in relationship with Christ, then they are both reverent and obedient to Christ Who is the Way, the Truth, and the Life. Christ lives in us and also in our spouse. We can understand mutual love more perfectly then as mutual obedience to the specific ways that Christ manifests the way, the truth, the life, and love in our spouse.
Next, St. Paul says that wives should be subordinate to their husbands. However he adds "as to the Lord." In a wife's personal relationship with Christ, she is faithful to him, loving, subordinate in the sense of obedience to whatever Truth and Love Christ teaches her. In the same way, a wife should be loving and faithful to her husband, obedient to whatever truths or insights he brings to the marriage as one in whom Christ lives.
How often in a deep discussion or argument, does a wife stay stubborn and refuse to listen to the truth or wisdom her husband has to offer about a particular topic? Yet, because we are members of Christ's Body, Christ dwells in each of us. If the husband, inspired by Christ, is right on this matter, the wife needs to stop being stubborn, selfish, or immature, and defer to her husband's wisdom and insight. That's being obedient to Christ the Truth.
What was a totally new insight for the ancient world is that St. Paul says "Husbands, love your wives, even as Christ loved the Church and handed himself over to her to sanctify her....So husbands should love their wives as their own bodies." Husbands in Paul's day thought of their wives as their property. And - we still see many men operating in that way today! Domestic violence is rampant. Many men still want to live out of that old hierarchical model that allows them to be the Boss in everything.
But for a man to love a woman out of reverence for Christ is for a man to admit that Christ lives in his wife, and gives her matchless dignity. She is worthy of respect. If a man loves his wife as Christ loved the Church, he has to be willing to live and to die for her. He hands himself over to her - he is willing to be obedient to whatever truths and insights that she brings to the marriage because Christ the Way the Truth and the Life lives in her. If he loves her as his own flesh, he will never strike her with his hand or in a verbal assault.
If both husband and wife are subordinate/obedient to each other, there can be such freedom, joy, and creativity in a marriage! Each can be independent in using his/her own matchless gifts because his/her spouse is obedient to the truth of who his/her spouse is, sees the gifts and affirms them and encourages them.
Both, as members of Christ's Body, know that the eye can't do what the ear does and the ear can't do what the hand does. So they are both also dependent on each other so that each can bring his/her own wisdom and insight and talent and truth and love to the marriage. Neither one is threatened by the other. Neither one "lords it over" the other. They rejoice in each others' differences and are twice as strong as a couple by pooling their strengths.
When a couple is subordinate to one another they find Christ the Way in each other. Each has a piece of the puzzle of how to proceed forward in hope and trust that their marriage can succeed, following Christ. They find Christ the Truth in each other, who reveals to each of them the ways in which they are selfish, immature, not totally committed. They find Christ the Life Who encourages them to bring new life into the world.
So - should wives be submissive to their husbands? Yes - and husbands should be submissive to their wives. Both should love and reverence and trust each other through Christ, with Christ, and in Christ. He will show them both the Way to a stable, happy, fruitful marriage. He will give them the humility to be obedient to the Truth that each one brings to every situation. And in Him, they will find happiness and abundant Life!