The Bible does not address the issue of birth control so it is difficult to say whether or not it is right or wrong in an absolute sense. But since it doesn't condemn it outright, we do not either.
Biblically, having children is a blessing from God and a fulfillment of the command of God to multiply and fill the earth (Gen. 1:28). If anything, we are told to have children, and never told to avoid having them. But that alone isn't enough to settle the issue. A generic principle of having children is for humanity and not always for individuals. Take Jesus, for example. He did not marry and have children. He did not fulfill the command to do so. Therefore, since Jesus never sinned, we can conclude that the command to have children is a generic one for mankind and not necessarily intended to be applied to each and every individual. Therefore, it would seem logical that birth control would be permissible to some degree.
If a couple decides to practice birth control, they must take special care that the method used is not abortive. That means that birth control is not acceptable if it causes a fertilized egg, fetus, or baby to be aborted. This is simply not right since we are told not to murder (Exodus 20:13). Remember, the life in the womb is human by nature and we do not have the right to arbitrarily destroy it as a means of birth control. Nevertheless, we see birth control as biblically permissible and sometimes even wise -- given certain circumstances. Let's look at some examples.
Let's say that there is a woman with a congenital illness that can adversely manifest during pregnancy and potentially or probably kill her. Should she risk having children knowing that a pregnancy might take her life? The choice would be hers and I would not condemn her for abstaining from getting pregnant. But, what if she already had children and due to her age, the next pregnancy carries a heightened risk of killing her and the baby. What then? Is birth control appropriate? I would think so since it potentially preserves the life of the mother and ensures that her children will have her around to care for them. In this case it would be wise to exercise birth control.
Of course, some would say that this is not trusting God and that it is taking matters into her own hands. I think this is fallacious logic. Should we not have a surgery to fix a heart problem in a newborn baby because it means taking things into our own hands? After all, isn't this the way God made the baby? But some will say, no, God doesn't make anyone with defects. Like or not, the Bible disagrees. "And the Lord said to him, Who has made man's mouth? Or who makes him dumb or deaf, or seeing or blind? Is it not I, the Lord?" (Exodus 4:11). Aside from the obvious implications of that verse, if God forms us in the womb and sometimes allows defects to occur, then should we, so to speak, go against His work? This is a tricky question, but we can safely say that God has given us wisdom to heal illnesses. Therefore, we should seek to prolong and improve life and if this means preventing conception, then I think it is permissible.
Should couples not have children because of financial hardship? This is a difficult question to answer because we have to consider that people have different financial situations. God says He will provide for our needs and I believe He would provide for the needs of the children should financially strapped couples decide to have them. On the other hand, if a couple is barely able to feed themselves and their economic future is bleak, they should pray and seek to exercise wisdom concerning the timing of having children. They are responsible for their children and if they cannot care for them, then birth control might be preferable until the situation changes.
This brings up another issue. What if a Christian couple lives where there is a famine and people are dying daily? Should they then intentionally try and have additional children or should they wait until the danger passes? I would think wisdom would compel them to wait.
Since the Bible does not address the issue of birth control and because situations for couples are different, I believe that birth control is not a sin -- provided it is not abortive, and provide that the motives to do so are wise and biblical.