It is not a direct proportionality, since there are many other factors as well. But yes, the pattern is such that the richer countries, in general, have lower crime/violence rates.
Of course, there are rich bastards as well. There are billionaires using their money to sell weapons to African rebels. There are countries in which the easiest way to get rich is to walk over your competitors' corpses. But these are a bit different kinds of crime. If we are talking solely about crime and violence among common citizens/immigrants, I tend to answer your questions with "yes".
It is worth noting though that even in the poorest countries only a small minority actually turns to crime. I used to be EXTREMELY poor (we are talking about income of two digits in $$ a month), and not a single time a thought to turn to even a minor crime has crossed my mind. Even in the counties with highest crime rates in the world you are unlikely to get into trouble, if you know what you are doing. My former home country had a staggering crime rate, but I myself was attacked only once, and that was because of my stupidity: I chose the darkest alley there was to choose, and when I noticed a few shady persons eyeing me strangely, I thought they were just bystanders and didn't flee in time.