A hadith found in three places in Sahih al-Bukhari suggests that birth control is not recommended in Islam:
We asked (the Prophet PBUH) about it and he said, 'There is no harm if you do not do [coitus interruptus1], for if any soul (till the Day of Resurrection) is predestined to exist, it will exist."Sahih al-Bukhari 4138
First, this hadith does not say that birth control is bad. It merely say that there is no harm in not using birth control.
This hadith, however, is not a very high-quality one due to its meager chain of narrators. Below is a diagram of all of the chains of the authentic versions of this hadith:
This hadith has only an 11.79% chance of authenticity, which is quite low. In my mathematical hadith verification methodology (which I discuss here), a hadith that falls between 10% and 20% is munkar (strange and unlikely to be true, but not clearly false and fabricated). A munkar hadith is not strong enough to act as a basis for establishing sunna (the tradition of the Prophet PBUH). For this reason, we cannot say with certainty that a negative view of birth control is part of Islam. In this case, the commonsense view should be used that birth control is a matter of personal choice.