No. In themselves, prayer in school and other rituals have no impact. Your more general question, however, may have some basis in reality. Since a desire for a reward and fear of punishment often motivates behavior, a general belief that there is no judgment day may lead to increased immorality and crime. But this is not necessarily the case. Europe on average is more secular than the United States, but by many measures of morality, it is more ethical and less crime-ridden. Ultimately, it comes down to values, and a person does not need God to have values. Behaviors are not moral because God has ordained them as such. Instead, if a perfect God exists, He is perfect because He has chosen to align himself with what is moral and just, and these values would be moral and just whether He was there or not.