Do baby monitors keep babies safe?

No. Baby monitors only help you keep your baby safe. It is your care in not leaving your baby unattended that keeps them safe.
Nope. It is important to know that monitors have not been shown to reduce risk of sids in babies. Turning on or off your monitor will have no impact on your babies sids risk. Risk reduction for sids includes breastfeeding, back sleeping in a "relatively" cool room (68 degrees) without a hat. Your baby's sids risk takes a steep dive at 6 months of life. Put the money in your baby's college fund.
No. Over a couple of decades the use of various monitors have assisted in treatment of infant apnea & heart rate issues. However, they have not been shown to reduce the frequency of sudden infant death syndrome. Proper use assists us in keeping babies healthy, but the vigilance of parents cannot be replaced by these devices.