I approached this New Republic article (free registration required) cautiously. The lead for the article is "Proponents of strict "covenant marriages" want to end divorce, not save marriage. They'll accomplish neither."
My caution was justified. The author expresses a smarmy attitude towards the presumably evangelical proponents of covenant marriages, and mockingly quotes Arkansas Governor Mike Huckabee saying that "marriage works best when the people trying to make the institution work are in the hand of the Creator," as if that were a false statement.
Society has traveled a long way (for better or worse) from the first no-fault divorce to our current state. Marriage is in the state it's in precisely because of a lack of respect for the institution - a problem partially caused and exacerbated by divorce. The covenant marriage movement addresses this directly by promoting the truth of what marriage actually is and encouraging them to live it out in its fulness.
I would understand the author's skepticism if he provided some objective basis for it. If, say, couples were being poorly prepared for covenant marriages, or something like that. But there is nothing. The author seems to be against the idea of encouraging couples to accept a higher standard for their marriages. He also speaks favorably of raising the bar for entry into marriage by mandating longer waiting periods and greater preparation or by raising minimum ages for marriage. That's all fine and good, but it's no basis to object to couples setting a higher standard for themselves.
This generation of young people has experienced the hardship of broken families like no previous one. This is bad of course, but it is also leading to a renewal of marriage and family life. The author errs in seeing covenant marriage not as an attempt to accomplish this renewal by encouraging couples to live fully the truth of marriage as a covenant, but rather as a way "to force people to stay married."
Courageous couples entering into marriage with the intention of permanence, with the intention of putting the procreation in God's hands, and with the intention of sacrificing themselves daily out of love for Christ and each other can only have beneficial effects on society. Undoubtedly couples will fail - that is the human story, isn't it? But couples will also succeed, and that is the real story. Those families who do live out the call will transform the culture (ahem, see quote at the top of this page) by radiating joy and love, but also through their daily hidden sacrifices.
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P.S. All that said, I'm a little uneasy about states legalizing a separate marriage status for covenant marriages, because then what becomes of the "regular-marriage?" It seems to me to be ceding ground that shouldn't be relinquished. Couples who wish to make their commitment more solemn can do it without a separate category. I would also be interested in the kind of preparation is provided to couples entering into the covenant marriage.
Additionally, from a Catholic perspective, I wonder what the existence of a covenant marriage option does to the annulment proceedings.

A "Covenant" marriage would likely have little effect at all.
I was in a marriage wherein we both professed that we were covenanted to each other until the going got very difficult and my wife left me for another man but managed to make it appear otherwise.
Now, fifteen years later I remain faithful to my vows and my wife and her lover(the same one she left me for) are accepted in the Catholic Church publically as a married couple, even though the Roman Rota has twice upheld our Sacrament.
It was a waste of time for me to stand up for our Sacrament. That is why I no longer practice the faith passed to me by my mother, God rest her wonderful soul.
The Catholic Church, in spite of what it says in public, knows that marriages are being destroyed
through the actions of its tribunal system and through the actions of many priests who encourage annulments after encouraging divorces because they do not want to invest themselves in the hard work of making a marriage work. They also know that if an annulment case is heard entirely in the U.S. there is a greater than 90% chance that the outcome will favor nullity.
The Catholic Church is a sham. May God save her from herself and may God save faithful spouses who are raped by their spouses and the Catholic Church.