An interesting 2011 paper looks at crime rates since 1970 in the United States and 8 major European countries. The authors, mostly Italian, come to a conclusion that will surprise many people: Europe has become more dangerous than the United States:
In 1970 the aggregate crime rate in the seven European countries we consider was 63% of the corresponding US ﬁgure, but by 2007 it was 85% higher than in the United States. This striking reversal results from a steady increase in the total crime rate in Europe during the last 40 years, and the decline in the US rate after 1990. The reversal of misfortunes is also observed for property and violent crimes.
A few charts:
An important caveat is that these numbers exclude homicide. The US homicide rate is currently 3-4 times higher than in most European countries. As I've pointed out, this fact is due mostly to two factors: the extremely high rate of black-on-black homicide in the US (52% of all persons arrested in the USA for homicide are black), and of course the wide prevalence of guns in the USA.
Homicide is actually not terribly relevant to public safety. It's much more rare than all other violent crimes, and is overwhelmingly concentrated among certain subgroups. Most homicides occur within an existing relationship, and many others occur among criminal subgroups such as gangs or drug users. The chance of an ordinary European or American being murdered by a stranger in a crime of opportunity is infinitesimally small.
As for general background violence in society, Europe is, statistically, more dangerous. It's interesting to speculate about why this might be. I suspect mass hooligan confrontations between football fans probably plays some rule: Every weekend there are dozens of unruly confrontations between rival football fans which may generate dozens of arrests at once. But still, these have been going on for quite a while.
The authors of the study perform statistical analyses to try to determine why European crime has increased. They do not identify immigration as a significant factor, although they say this is mainly for lack of data. The one factor they do identify as significant is length of incarceration. They argue that Europe's comparatively lenient criminal-sentencing regimes help to explain the crime increase. They find that length of criminal sentence does have an effect on crime rates, and suggest that Europe should increase prison sentences.
At the end of the day, the universal rule for all developed societies holds: crime is concentrated among poor and minority areas, and if you avoid these, your chances of being the victim of a violent crime are minimal. But still, anyone who praises Europe as safer than the USA needs to update their stereotypes.