Claiming that "crime" (presumably violent crime, since that's the sort of crime in which banning assault weapons would matter) did not go down in the period between 1994 and 2004 (the time of the Federal Assault Weapons Ban, the "Clinton ban") is flat out wrong. That's easily seen by looking on the FBI or DOJ websites, because they freely host tables of crime statistics.
Comparing the FBI's listing of the violent crime rate between the two time points, we find that the violent crime rate in 1994 was 713.6, and by 2004 it was 463.2. That is definitely a motion in a downward direction.
So, yes, crime rates did go down between 1994 and 2004. However, this probably has very little to do with the Federal Assault Weapons Ban; looking at almost any graph of crime rates will show that general crime rates peaked in the early '90s before the ban, and have continued to fall even after the ban was lifted.