The controversy isn't over stem cell therapy, it is over embrionic stem cell therapy. Stem cell therapy usually refers to stem cells extracted from the patients own body. Embrionic stem cells are extracted from dead fetuses. Those opposed to embrionic stem cell therapies are worried that it will lead to 'fetus farms', where eggs are fertilized until a fetus is formed, then the fetus is killed to harvest the stem cells. It is an offshoot of the abortion controversy in which the pro-life side largely believes that life begins at conception (egg fertilization).
The controversy around stem cell therapy lies in the research in embryonic stem cells. Some people believe that the start of the embryonic stage is when a human being begins and is therefore unethical to destroy this for the stem cells as it is supposedly akin to murder.