Is the Catholic Church NOT against all birth control?

All forms of birth control are condemned.

Pope Pius XI, Casti Connubii (#’s 53-56), Dec. 31, 1930:

“And now, Venerable Brethren, we shall explain in detail the evils opposed to each of the benefits of matrimony. First consideration is due to the offspring, which many have the boldness to call the disagreeable burden of matrimony and which they say is to be carefully avoided by married people not through virtuous continence (which Christian law permits in matrimony when both parties consent) but by frustrating the marriage act. Some justify this criminal abuse on the ground that they are weary of children and wish to gratify their desires without their consequent burden. Others say that they cannot on the one hand remain continent nor on the other can they have children because of the difficulties whether on the part of the mother or on the part of the family circumstances. “But no reason, however grave, may be put forward by which anything intrinsically against nature may become conformable to nature and morally good. Since, therefore, the conjugal act is destined primarily by nature for the begetting of children, those who in exercising it deliberately frustrate its natural powers and purpose sin against nature and commit a deed which is shameful and intrinsically vicious. “Small wonder, therefore, if Holy Writ bears witness that the Divine Majesty regards with greatest detestation this horrible crime and at times has punished it with death. As St. Augustine notes, ‘Intercourse even with one’s legitimate wife is unlawful and wicked where the conception of offspring is prevented.’ Onan, the son of Judah, did this and the Lord killed him for it (Gen. 38:8-10).
“Since, therefore, openly departing from the uninterrupted Christian tradition some recently have judged it possible solemnly to declare another doctrine regarding this question, the Catholic Church, to whom God has entrusted the defense of the integrity and purity of morals, standing erect in the midst of the moral ruin which surrounds her, in order that she may preserve the chastity of the nuptial union from being defiled by this foul stain, raises her voice in token of her divine ambassadorship and through Our mouth proclaims anew: any use whatsoever of matrimony exercised in such a way that the act is deliberately frustrated in its natural power to generate life is an offence against the law of God and of nature, and those who indulge in such are branded with the guilt of a grave sin.”

How does this square with Benedict XVI's promotion of condoms or NFP? Of course it doesn't.

In his book, Light of the World pages 117-119, Benedict XVI states the following:

Question: “Are you saying, then, that the Catholic Church is actually not opposed in principle to the use of condoms?

Answer from Benedict XVI:

“She of course does not regard it as a real or moral solution, but, in this or that case, there can be nonetheless, in the intention of reducing the risk of infection, a first step in a movement towards a different way, a more human way, of living sexuality

The two requirements of sexual intercourse within marriage are:

1. Procreation (to make babies)
2. Unity (union with spouse)

On the first, the Church teaches that the act must be "ordered to" procreate. Meaning that there ought to be no artificial impediment to the act of procreating. They also say that the act should be "open to life"

On the second point, the couple should have physical intimacy. (This rules out things like artificial insemination as misuse of sexuality.)

The Church has authorized a method called NFP (Natural Family Planning) which basically means you can time having sex so that you don't have kids. However, even NFP can't just be used in any circumstance because you don't want to have kids, as noted in the Catechism:

2368 A particular aspect of this responsibility concerns the regulation of procreation. For just reasons, spouses may wish to space the births of their children. It is their duty to make certain that their desire is not motivated by selfishness but is in conformity with the generosity appropriate to responsible parenthood. Moreover, they should conform their behavior to the objective criteria of morality:

When it is a question of harmonizing married love with the responsible transmission of life, the morality of the behavior does not depend on sincere intention and evaluation of motives alone; but it must be determined by objective criteria, criteria drawn from the nature of the person and his acts criteria that respect the total meaning of mutual self-giving and human procreation in the context of true love; this is possible only if the virtue of married chastity is practiced with sincerity of heart.

Thus given the context for your question was a discussion about people in poverty, the Cardinal was talking about people in need being able to control their family size through NFP. Perhaps there is a deeper philosophical point he's trying to make that the Church doesn't categorically just say "no birth control", but NFP is really the only method we know of that meets the criteria.

The above is perhaps better explained by Humanae Vitae (Pope Paul VI, 1968). Note that it talks about:

1. Not using birth control to prevent conception
2. That these inanimate objects are not evil in themselves (i.e. a woman could use what is essentially a birth control pill if the objective of that pill is therapeutic for a reason other than preventing birth)
3. That you can space out births by timing sex.

Unlawful Birth Control Methods

14. Therefore We base Our words on the first principles of a human and Christian doctrine of marriage when We are obliged once more to declare that the direct interruption of the generative process already begun and, above all, all direct abortion, even for therapeutic reasons, are to be absolutely excluded as lawful means of regulating the number of children. (14) Equally to be condemned, as the magisterium of the Church has affirmed on many occasions, is direct sterilization, whether of the man or of the woman, whether permanent or temporary. (15)

Similarly excluded is any action which either before, at the moment of, or after sexual intercourse, is specifically intended to prevent procreation—whether as an end or as a means. (16)

Neither is it valid to argue, as a justification for sexual intercourse which is deliberately contraceptive, that a lesser evil is to be preferred to a greater one, or that such intercourse would merge with procreative acts of past and future to form a single entity, and so be qualified by exactly the same moral goodness as these. Though it is true that sometimes it is lawful to tolerate a lesser moral evil in order to avoid a greater evil or in order to promote a greater good," it is never lawful, even for the gravest reasons, to do evil that good may come of it (18)—in other words, to intend directly something which of its very nature contradicts the moral order, and which must therefore be judged unworthy of man, even though the intention is to protect or promote the welfare of an individual, of a family or of society in general. Consequently, it is a serious error to think that a whole married life of otherwise normal relations can justify sexual intercourse which is deliberately contraceptive and so intrinsically wrong.

Lawful Therapeutic Means

15. On the other hand, the Church does not consider at all illicit the use of those therapeutic means necessary to cure bodily diseases, even if a foreseeable impediment to procreation should result there from—provided such impediment is not directly intended for any motive whatsoever. (19)

Recourse to Infertile Periods

16. Now as We noted earlier (no. 3), some people today raise the objection against this particular doctrine of the Church concerning the moral laws governing marriage, that human intelligence has both the right and responsibility to control those forces of irrational nature which come within its ambit and to direct them toward ends beneficial to man. Others ask on the same point whether it is not reasonable in so many cases to use artificial birth control if by so doing the harmony and peace of a family are better served and more suitable conditions are provided for the education of children already born. To this question We must give a clear reply. The Church is the first to praise and commend the application of human intelligence to an activity in which a rational creature such as man is so closely associated with his Creator. But she affirms that this must be done within the limits of the order of reality established by God.

If therefore there are well-grounded reasons for spacing births, arising from the physical or psychological condition of husband or wife, or from external circumstances, the Church teaches that married people may then take advantage of the natural cycles immanent in the reproductive system and engage in marital intercourse only during those times that are infertile, thus controlling birth in a way which does not in the least offend the moral principles which We have just explained. (20)

Neither the Church nor her doctrine is inconsistent when she considers it lawful for married people to take advantage of the infertile period but condemns as always unlawful the use of means which directly prevent conception, even when the reasons given for the later practice may appear to be upright and serious. In reality, these two cases are completely different. In the former the married couple rightly use a faculty provided them by nature. In the later they obstruct the natural development of the generative process. It cannot be denied that in each case the married couple, for acceptable reasons, are both perfectly clear in their intention to avoid children and wish to make sure that none will result. But it is equally true that it is exclusively in the former case that husband and wife are ready to abstain from intercourse during the fertile period as often as for reasonable motives the birth of another child is not desirable. And when the infertile period recurs, they use their married intimacy to express their mutual love and safeguard their fidelity toward one another. In doing this they certainly give proof of a true and authentic love.

(the full document: http://www.vatican.va/holy_father/paul_vi/encyclicals/documents/hf_p-vi_enc_25071968_humanae-vitae_en.html)