The case for Western-style marriage as a way to protect woman irrevocably relying on their spouse for economic support when having children
While this isn't a universal justification for marriage, it is an economic reason for marriage that heavily drives the decision to marry in modern times in the developed world in a large share of cases.
This analysis ignores what have historically been uncommon edge cases of couples who by choice or due to infertility don't have children, who are simply swept along following marriage rules that weren't really designed for them, even though they are now far more common, often for reasons that are not primarily economic, or coincidentally benefit from marriage rules not really designed for them.
The explanation of marriage on this economic basis also explains the class divide between marriage utilization in modern societies, with marriage remaining a strong institution for the affluent, while it falls apart from the working class and poor people.
In modern circumstances
Stereotypically, and very frequently in practice, when a man and a woman marry, they have children and the woman ends (temporarily or permanently) or scales back, her participation in the paid workforce. Particularly in the case of woman who have high levels of education or specialized training, doing so dramatically reduces the woman's lifetime earnings. This factor is much less important for woman with little education and/or few specialized skills who pay a much smaller penalty in their lifetime earnings for temporarily leaving the workforce or reducing their earnings while their children are young.
So, for it to make sense for a woman to have her male partner's children with the economic sacrifices associated with that decision, she needs legally enforceable protections for the reliance she places on her male partner to economically support her.
If the relationship falls apart, in the absence of marriage, the only remedy a woman who has sacrificed her lifetime earnings to have children, is child support, which only partially compensates the woman for the cost of raising the child that she had with her co-parent on a part-time basis (as the co-parent usually has some custody rights).
But if she is married and the relationship falls apart, she is entitled not only to child support, but also to a division of marital property and often substantial alimony (a.k.a. maintenance), which reflects her sacrifice of lifetime earnings in her name.
Of course, this remedy only has value (both as a remedy if the marriage ends, and as an incentive for her husband to not divorce her in the first place), if the husband has meaningful assets from which there will be marital property to divide and/or meaningful income from which alimony and child support can be paid.
These considerations help to explain why, in the modern U.S., college educated woman who are married to college educated men or non-college educated men with good economic circumstances, are much more likely to marry before having children (the average is two years after marriage), and are much less likely to divorce, than less educated and/or less economically skilled women.
In contrast, women who have relationships that could produce children with men who are not college educated and have poor economic circumstances tend not to marry before having children (the average is two years before marriage if the couple marries at all), are more likely to divorce if they have gotten married, and are less likely to even acknowledge their co-parent on a birth certificate, because in these circumstances, marriage does not provide meaningful economic protections to the woman having children and when the father is particularly destitute or potentially troublesome, even acknowledging the father's paternity may result in her paying child support to the father and being forced to continue to have him in her life.
In "pre-modern" circumstances
In pre-modern times (i.e. when there were far fewer economic opportunities for single mothers, and contraception outside marriage was unavailable or unreliable, and women tended to have far more children during a lifetime, so really in the 1950s and earlier), failure to have the complete economic support of a husband meant that a woman without a husband and any children she had living with her, were likely to experience extreme economic privation. Very few women had the ability to economically support themselves and their children without a husband in this era (only wealthy women who had significant income generating property could manage this feat).
Also, for most of that era, the state lacked a meaningful capacity to enforce child support, alimony, or shared custody arrangements. So, in that era, a blanket prohibition on divorce absent the most extreme circumstances (and usually even then restricted people who could survive on the property that they owned alone), was the only way to legally protect a woman's reliance on her co-parent to economically support her and their children.
This extreme economic imperative overrode modern era concerns about a relationship falling apart and a couple being forced to live in a "dead" marriage.
Alternatives to modern Western-style marriage and the economics driving these alternatives
A modern Western-style marriage isn't the only possible solution to the problem of providing economic security that allows women to feel comfortable making the economic sacrifices involved in having children.
Matrilineal societies with mothers and a male relative heading a household
There are some societies where women have children outside of marriage, sometimes with many men having children who are half-siblings. In those societies, mothers frequently establish households in a non-sexual pairing with a brother or uncle or father. But, most societies historically, and most modern societies, in particular, have not made that choice.
Strong social safety nets that make single motherhood viable
In some countries with strong social safety nets and third-party child care systems, especially in Scandinavia and Western Europe, but also in some countries with Soviet-style communism, single motherhood was much more economically viable and marriage was less common, because yet another solution to the problem of economic support for single mothers was devised.
Polygyny and removing some men and women from parenthood
Historically, in societies where a large share of men lack the economic capacity to support a woman and her children, while a minority of men are affluent, polygyny (sometimes with equal wives, and sometimes with a primary wife and secondary wives or concubines or slaves) solves the economic problem. Other solutions to that situation have been for a certain percentage of women to be celibate in holy orders or as priestesses and to forgo having children. Societies with this economic make up also sometimes have significant portions of men who are eunuchs and/or who become celibate and pursue religious callings.
Polyandry
Historically, in societies where almost all men lack the economic capacity to support a woman and her children alone, polyandry (often with two brothers marrying a single wife) is known and solves the economic problem.
Comment: Do you have any comment on how civil union or domestic partnership arrangements differ from marriage, or are they just marriage by a different name for legal purposes? – Ian Sudbery
Response: In the U.S., civil unions were invented to be marriage for same sex couples when it was politically unpopular and not constitutionally required to allow same sex marriage. Institutions like PACS (Pacte Civil de Solidarité) en.wikipedia.org/wiki/Civil_solidarity_pact in France were motivated by the purpose as well, but became popular with unmarried couples who wanted some but fewer legal implications of marriage, often childless cohabiting couples, or couples who didn't want the cultural/religious baggage that the predominant Roman Catholic church there assigns to marriage. – ohwilleke
