Generally speaking, Christians are pro-life. If prenatal testing can help benefit the life and health of the coming baby, isn't that in their best interest? Most Christians would choose to keep the baby, even if it showed signs of disease or deformity... Every parent wants the best for their child, and that includes Christians.
Christians should pursue pre-natal testing. All pre-natal testing does is provide knowledge. What Christians might choose to do or not to do once they get that knowledge, is up to themselves, and only themselves (not an entire religion), but gathering information that is personal and directly related to one's life is certainly acceptable, without argument.
The only reason I can see why they shouldn't is because they are pro-life, but that is also a reason why they should. Prenatal testing is helpful for some in deciding if they want to abort a child, but that is far from the only reason to do it. The medical field is able to do a lot in the form of prenatal health care. They can even do surgeries while the child is still in the womb. Christians who have genetic problems in their family ought to know if their child is carrying those genes so that they can be prepared for problems before the child is born.
When couples are expecting a child, they are sometimes asked by friends and colleagues if they are going to have prenatal testing on their child. Why would they ask such a question? The purpose of prenatal testing is to screen for any genetic defects or diseases which the unborn child may possess.
The unspoken corollary to this purpose is that couples may wish to abort a child with genetic disorders such as Downs Syndrome. To pursue prenatal testing is to presuppose the contraceptive mentality which is unacceptable for Christians. Christian couples should be open to the gift of life, no matter what kind of life they have conceived.