The Church “appreciates and encourages the progress of the biomedical sciences which open up unprecedented therapeutic prospects”  . Such research, however, must be conducted with respect for human life and the dignity of the person.
There is no ethical problem with obtaining stem cells from umbilical cord blood, placenta or adult tissues since they can can be acquired without harming the donor. These types of stem cells are generally called "adult stem cells." Another kind of stem cell research involves the destruction of embryonic humans in the first weeks of their development. Such research requires the "deliberate killing of innocent human beings," for this reason it is a grave crime against life. 
The USCCB's June 2008 Statement on Embryonic Stem Cell Research outlines three responses to common arguments in favor of embryonic stem cell research. Below is a brief summary of these arguments and the USCCB response.
The benefits outweigh the harm done to the embryo.
Evil can not be justified for the sake of a good end. A desire for a "greater good" does not take away from the evil of deliberately killing an innocent human person.
The embryo at this stage is not a human being with full rights.
It is a scientific fact that from conception the embryo is alive, has complete human DNA and "is actively expressing those genes to live and develop in a way that is unique to human beings, setting the essential foundation for further development." The dependence of the embryo does not diminish the fact that he or she is a "complete and distinct member of the species Homo sapiens." [ 3]
The fact that at this stage the embryo lacks certain mental and physical abilities does not take away from the human dignity and value that is inherent in every human. If the fundamental value and rights of the person is dependent on his or her abilities or qualities that can "grow or diminish, and be greater or lesser in different human beings, then there are no inherent human rights, no true human equality, only privileges for the strong." 
Dissecting embryos "should not be seen as involving a loss of embryonic life," because the embryos are unwanted or will die anyway.
All of us will eventually die, but this does not give anyone the right to kill us. 
1 Pope Benedict XVI, Address of January 31, 2008
2 On Embryonic Stem Cell Research, USCCB, June 2008
3 On Embryonic Stem Cell Research, USCCB, June 2008
4 On Embryonic Stem Cell Research, USCCB, June 2008
5 On Embryonic Stem Cell Research, USCCB, June 2008