Start with googling "did crime fall during the great depression" because reputable sources will show you that no, during the first part it did not and then it only fell after recovery programs began to be put in place to put people to work. As well as noting there is going to be more correlation with property crime and poverty than with violent crime. You also have to consider confounding factors such as that in cities crime went up during Prohibition and then back down when Prohibition was repealed in 1933 which overlaps with Great Depression years.