So America has clearly become more secular in the last 40 years. What has crime done during this time? The data indicates that since 1970, crime has increased, and then decreased (see the chart below that I made in Microsoft Excel, using data from the BJS).
In 2007, violent crime was roughly where it was forty years earlier. And property crime was actually below where it had been. Decreasing religious belief has either had no impact, or a slightly positive impact, on the American crime rate.
God and Crime Around the World
As a general rule, religiosity is highest among the poorest nations of the world, and lowest among the richest. This Gallup study of global religiosity, asked people around the world if religion was an important part of their daily lives. Unsurprisingly, there is a general correlation between the importance of religion in daily life, and homicide rates.
Less scientifically, we can generally see that the more secular and agnostic societies of the rich world tend to be among the most peaceful, at least in the area of homicides. The US is the most murderous country in the rich world by far, while being one of the most religious. About 33% of Americans indicate religion is not important in their daily lives. For the least murderous societies (which include Singapore, Austria, Norway, Switzerland and Germany), this number is never lower than 40%, and goes as high as 78%.