Typically, the most prominent example of how "men should have authority over women" stems from Ephesians 5:22, in which Paul admonishes wives to submit to their husbands.
That said, a contextual reading will bear much fruit:
21 submitting to one another out of reverence for Christ.
22 Wives, submit to your own husbands, as to the Lord. 23 For the husband is the head of the wife even as Christ is the head of the church, his body, and is himself its Savior. 24 Now as the church submits to Christ, so also wives should submit in everything to their husbands.
25 Husbands, love your wives, as Christ loved the church and gave himself up for her, 26 that he might sanctify her, having cleansed her by the washing of water with the word, 27 so that he might present the church to himself in splendor, without spot or wrinkle or any such thing, that she might be holy and without blemish.[a] 28 In the same way husbands should love their wives as their own bodies. He who loves his wife loves himself. 29 For no one ever hated his own flesh, but nourishes and cherishes it, just as Christ does the church, 30 because we are members of his body. 31 “Therefore a man shall leave his father and mother and hold fast to his wife, and the two shall become one flesh.” 32 This mystery is profound, and I am saying that it refers to Christ and the church. 33 However, let each one of you love his wife as himself, and let the wife see that she respects her husband.
In a nutshell, a few facts are pretty plain:
Husbands are to love their wives
Verse 21 & 22 can be read separately or independently
If one is going to base a headship claim in 5:23, then one cannot ignore 5:24 - 25 without being seriously inconsistent.
Traditionally, 5:22 is often read "Wives, submit to your husbands." Note, this is not a blanket invitation to tyranny, however, as there is clearly an equal and separate obligation to the husband to love and care for his wife.
This has, especially as society at large has changed, led many theologians to read verse 21, not as a separate sentence, but as an introductory clause. In the Greek, as there is no punctuation, it is this equally possible that the text reads as follows:
Submitting to one another out of reverence for Christ, wives, submit to your own husbands, as to the Lord.
This concept of mutual submission in love also squares well with the general concept of all being equal before a sovereign God.
Note, however, that an understanding of "headship" is by no means universal across denominations. Even within the Catholic Church, there is a wide spectrum of understanding what exactly the rights and responsibilities of authority entail.
Regardless of the exact nature of authority, however, it is also important to understand that any right of authority is, by definition to be tempered with love. While it would be easy to point out that Ephesians 6 picks up the theme saying, "Fathers, do not exasperate your children," I'd suggest rather that there is more fruit to be had in looking at what love means.
Love, in 1 Corinthians 13, is pretty straightforward:
Love is patient, love is kind. It does not envy, it does not boast, it is not proud. 5 It does not dishonor others, it is not self-seeking, it is not easily angered, it keeps no record of wrongs.
Even if one chooses to equate headship with authority, the fact that it is to be exercised in love should naturally proscribed certain abuses inconsistent with love in any event.