The only serious approach to eliminating crime is to eliminate its very grounds, to make it worthless. If one can reach well-being by robbing the others, one is tempted to commit any crime, and hence one is ready to become a criminal, under favorable conditions. On the contrary, if no criminal act lead to any profit, there would be no motive to commit it.
The absolute majority of crimes aim at re-distribution of wealth, when one person wants to seize what belongs to another person, by any means, up to murder. Traditionally, will for power takes is second line of the chart, but it is usually a mere derivative of money. If there is no property at all and no profit from power, what can murder change? Even assuming that criminals are mentally insane, the elimination of the mad rush for gain would significantly reduce the risk of getting mad.
Obviously, the ruling classes will hardly ever think about stopping the crime as such, and all they do is neutralizing individual criminals, which only gives way to other criminals, and thus to infinity. This is the well-known game of cops and thieves, described by Eric Berne.
The culture based on property, under the name of civilization, starts with crime and ends with crime; crime is in its very core. The first act of appropriation is already robbery, since one person pretends to have exclusive rights to what belongs to the whole humanity.
No wonder that the modern capitalist society is replete with crime, essentially depending on it. Capitalism is interested in reproducing the atmosphere of crime, and that is why all the mass media, all the entertainment industry tries hard to persuade people that nothing can be more attractive than crime—except, probably, sex; and a mix of the two is a real blockbuster. Slaves must live in fear, to be more obedient. Those at power put forth the myth of "legality" as a social system that prevents the evil from invading people's life, and the guards of the capital pretend to be defending the well-being of everybody. People are made to believe that the destruction of that "protective" system would result in chaos, and that is why the existing society, however poorly organized, should be kept on just for the sake of stability...
Of course, if one assumes that there is no social organization other than capitalism its destruction would seem a catastrophe. For a bourgeois, who cannot imagine his existence without property, any attempt to eliminate property will look like a robbery. Elimination of bourgeois legality would ruin the "legal" right of one person to appropriate the work of another. When a criminal robs somebody, this in no way threatens the system, being the individual problem of that person, while any legal infringement of the sanctuary rights of the proprietor is a blow to the whole class of proprietors, and hence a much more serious crime in the eyes of an apologist of capitalism.