Has the Catholic Church or any large Christian organization ever had a policy of paying people to not have abortions?

This actually comes down to a perspective question, the facts and sources hardly matter, however lets start with a source, the critical one for answering questions regarding Catholicism The Catholic Catechism states:

2286 Scandal can be provoked by laws or institutions, by fashion or opinion.

Therefore, they are guilty of scandal who establish laws or social structures leading to the decline of morals and the corruption of religious practice, or to "social conditions that, intentionally or not, make Christian conduct and obedience to the Commandments difficult and practically impossible."88 This is also true of business leaders who make rules encouraging fraud, teachers who provoke their children to anger,89 or manipulators of public opinion who turn it away from moral values.

2287 Anyone who uses the power at his disposal in such a way that it leads others to do wrong becomes guilty of scandal and responsible for the evil that he has directly or indirectly encouraged. "Temptations to sin are sure to come; but woe to him by whom they come!"90

You may be confused as to how this relates, but deeper thought will lead us to understand why the Catholic Church does not provide condoms for the poor or in this case provide money to people who do not get abortions. The Catholic Church, despite having many branches that do, can never do or advocate these kinds of things at the higher level. They would be encouraging sin if they did. Someone who knows they can be paid to have a child may start conceiving more children in an immoral manner to get paid.

But, as David Stratton has demonstrated this can all be resolved at the local level. Say I personally know Jane, she has been a good person, but her boyfriend was a jerk and somehow convinced her to do something she now regrets. A general policy of the church not to get involved, is not going to stop me from getting involved as a good Catholic who understands we all make mistakes.

To be clear, I am not saying that even a fraction of Christians believe no help should be provided to such individuals. Just that the help should be given by those who understand the situation, not some overarching body of regulations. Christians of any denomination always believe in helping the poor, but unless they themselves know a good portion of the specifics around any situation, they cannot know the person is poor, needs help, or is even willing to accept help.

How would a Christian church respond if someone said they would kill unless they are paid by the church. I would hope the church refuses to pay, unless the church knows something about this threaten-er I don't. Because if they did pay, they would only be encouraging others to conduct the same actions.

John Cardinal O'Connor, archbishop of New York of 10-20 years ago, once declared that if any woman were having an abortion because of the financial difficulties, to come see him.

And as far as "paying the mother to carry child to term", this is essentially the terms of some adoptions. I know of a case where this happened, where an adoption was arranged specifically in response to a concern about risk of abortion.

This takes the long route to coming to the actual answer. Unfortunately, the ultimate answer is purely speculative. Whether any given entity would do something is all speculative compared to asking whether a given entity has done that thing. The best we can do is look at what has been done, what has been said, and use logic applied to the complexity and uncertainty of human behavior, and say, "based on the evidence this is the most likely answer." That's not an approach that will give us "the correct" answer, which is the typical StackExchange requirement, but I'll give it a go.

My approach will be to document what has been or is being done and draw the most likely conclusion based on that evidence.

Establishing an answer to whether the desire exists.

Evidence 1: Many Churches do pay for programs that help women in this situation. My own Church, with less than fifty adults old enough to work and earn money to give, partners with other area Churches to support a local organization called the Agape Center.

The Agape Center is founded, staffed, and financed by other like-minded Churches with no aid from anyone other than the members of these Churches. In essence, it is an extension of the Church for the Churches that have banded together to create this center.

The center offers information, counseling, support, and care for women with "crisis" pregnancies. We wish we could provide the type of financial aid you reference. We simply don't have the financial resources to do so, but we would if we could.

And we are not alone. Many Churches have similar outreaches. Some are better funded and are able to do more.

I honestly don't know of any Church or Church organization that wouldn't foot the bill to save lives, if it had the resources. I can't imagine a Church that would be so cold to those in need. It goes against basic, near-universal Christian principles of love and recognizing the value of life.

LDS, Catholic, Baptist, Lutheran, and many other Churches I've been involved with, attended, or otherwise dealt with have all had programs to help unfortunate women in this situation, and every single one would love to be able to do more.

Summing up the above, most Churches have a strong desire to do what they can to prevent and end abortions, and are willing to contribute financially.

The logical conclusion is, if they thought it were feasible I'm sure plenty of Churches would do it.

But that's pure speculation. So I'm marking this as a community wiki, expecting it to be deleted.