This is a tough question because it combines aspects of religion, politics, and science. Because of that, a person's opinions on it can often depend on which aspect they are looking at.
The pros come when you look at the benefits that might be achieved through the use of stem cells. Stem cell therapies hold the hope of curing conditions that cannot be cured right now. This would clearly be a huge benefit to people with those conditions and to their loved ones.
The cons are largely religious and ethical. Embryonic stem cells (the most promising kind) must be taken from human embryos. This would destroy the embryo. Depending on your religious/political/ethical views, this may be killing a human life. If you believe that life begins at conception, then you see this as killing one person to save another's life.
So what it comes down to is a debate over fundamental issues such as the definition of life and when life begins. Your position on that must define your position on the use of embryonic stem cells.