Christian Right Opposition to Birth Control:
Opposition to birth control is increasing in evangelical circles which are, curiously, relying heavily on Catholic teachings. While Catholic teaching officially rejects the use of contraceptives, you don't see Catholics trying to undermine access to them - but that's what some evangelicals seem to have in mind. This is strange given the traditional Protestant support for family planning, but it makes sense in the context of the Culture Wars and some Christian traditions.
Sex Education & Abstinence-Only Programs:
One of the most visible means which the Christian Right uses to attack birth control is by opposing education about contraceptives and contraceptive techniques in sexual education classes in the public schools. There is a big push for abstinence-only programs, where students are taught that only about abstinence - not physical or chemical contraceptives. This helps people grow up relatively ignorant: if they aren't aware of their options, then they aren't likely to choose contraception.
Advertising of Contraceptives:
For a long time, TV stations refused to run ads for any contraceptives at all. These restrictions have recently been lifted somewhat, but condoms still aren't advertised as a general rule. You also don't see ads for them on the web. Christian protestors are able to get public ads for contraceptives taken down if they are able to argue that even the suggestion of sexual activity qualifies as lewd, obscene, or somehow in violation of community standards.
Pharmacists & Conscience without Consequences:
An increasingly popular form of anti-choice activism is for pharmacists to refuse to dispense emergency contraception
(Plan B, Morning After Pills). It is argued that they cause abortions, even though the most they do is prevent implantation. Limiting access to emergency contraception keeps women from preventing pregnancies and increases the demand for abortion. It also helps blur the lines in between abortion and contraception, useful in any long-term effort to undermine contraceptive rights.
Availabilty of Contraceptives:
If Christian Right activists can ensure that pharmacists don't have to provide access to contraceptives, this will open the door to putting pressure on pharmacists to refuse to provide contraceptives no matter what they think. The situation for contraceptives would be analogous to the current one for abortion: legal, but providers labor under intense pressure to get out of the business. The Christian Right may not be able to make birth control illegal, but they may could make it unavailable.
Right to Privacy and Contraceptive Use:
If Americans have a right to bear children, don't they also have a right not to have children? It would be difficult to argue that there is a fundamental, constitutional right of privacy which encompasses a right to procreate but which doesn't cover a right not to procreate. Christian Right attacks on contraception are thus also attacks on people's privacy rights, as well as on their fundamental rights regarding whether or not they have children.
Sex, Consequences, and Contraception:
America's Protestant Christian Right has no objections to contraception per se and certainly no objections to the principle of family planning. What they object to is the use of contraception in order to allow people to engage in extramarital sex without worrying about the possible consequences, including pregnancy and sexually transmitted diseases. This is completely consistent with not only Christian traditions, but also traditions in other religions.
Why, though, do they object to preventing some of the consequences of extramarital sex but not marital sex? Why is it okay to prevent pregnancy within marriage but not with couples who are not married? That's a difficult question to answer, but it appears that it lies with the assumption of the essentially sinful nature of sexual acts. Traditional Christianity only permitted sex for the purpose of procreation; modern Christianity allows for sex having a purpose in marital harmony and as an expression of marital love.
Extramarital sex of course cannot legitimately partake of any of these goods, and is thus necessarily sinful. It's okay for non-sinful marital sex to occur without visible consequences, but people engaging in sinful extramarital sex should not be allowed to go unpunished. Pregnancy and disease are personal consequences of sex which can serve as a form of punishment and therefore should not be prevented.
The Christian Right believes that there are already widespread negative social consequences of unrestrained extramarital sex - hedonism, loss of respect for marriage, and so forth. If there were stronger and more obvious personal consequences to extramarital sex, then maybe fewer people would engage in it and the social consequences would be reduced.