Only at a low level. Changing a fine for stealing a loaf of bread to a five year jail sentence isn't going to stop someone stealing a loaf of bread if they are starving. Even the Saudi approach of chopping limbs off only works if the shopkeeper is minded to ensure that the thief is reported - I suspect many would just not report the theft instead or deal with it in other ways. Plus once you go down that route it ends up costing more in healthcare and inability to work.
US states that have downgraded shoplifting to a misdemeanour appear to have had a surge in shoplifting, so penalties are a deterrent. They're only a deterrent for crimes you think about though. Most violent crime is spontaneous and even then with certain premeditated crimes (rape, serial killers) the low risk of being caught is enough to outweigh the possible punishment.
When the US has the option of life without parole in a grim jail I don't know why they persevere with the death penalty, other than state revenge. From a purely objective angle, the logistics and economics of it don't add up.
