In 1973, Rabbi Moshe Tendler raised to his father-in-law the issue of testing for Tay-Sachs, a genetic disease found among Jews of Ashkenazic descent. The disease can only be contracted by a child whose parents were both carriers of the disease (even though it may not affect the parents at all), and the horrific impact is generally a very short life for the child. By the time of the question, reliable blood tests existed to help determine whether or not a person is a carrier for this disease, and Rav Moshe Feinstein was asked to rule on whether or not people should be tested before getting married.
Rav Moshe's initial response is to say that given the existence of such blood tests, avoiding having such a test would not only not be a lack of faith in God, but would actually be a case of closing one's eyes to something that he has the ability to know. Thus, Rav Moshe does encourage such blood tests to be taken by people who are contemplating marriage.
However, Rav Moshe does offer some caveats. First, he stresses that this should be a private process, to be revealed only when it is important with regard to a specific shidduch. Second, he notes that since people often get nervous about such issues and often make a bigger deal out of things than they should, therefore people should not be tested or even talked to about such testing until they are legitimately of an age when they may begin looking to get married.
Finally, it seems that there was a fund for Tay-Sachs research that included non-Jewish and non-religious doctors, which, among other things, encouraged the aborting of babies who were determined while in utero to have Tay-Sachs. Rav Moshe discouraged religious doctors from participating in this organization, but noted that if there were beneficial and halachically permissible aspects to this organization then they could join so long as they did not in any way lend their support to the aborting of babies.