Yes, I think that the absence of punishment or torture in anarcho-capitalist society would increase crime rates, because people will do terrible things if they think they can get away with them. Most people are basically good. But other people are not. People still have to deal with their bad nature, and that means that if given the option, people will turn into Lord of the Flies.
The absence of punishment or torture in an anarchic-capitalist society would increase crime rates. People and criminals have to be afraid of something and doing away with any type of punishment would just cause the crime rate to increase. It would give criminals a way out of any type of harsh punishment.
Many advocates of anarcho-capitalism would seek to abolish forms of punishment such as imprisonment and replace it with a system of financial fines instead. The vast majority of those who commit crimes however, are at the bottom of society with no finances to begin with. Thus fines are of little to no deterrent. Furthermore are habitual criminals completely incapable of self control and/or empathy and will commit crimes until physically halted from doing so. In a system with no prison or punishment they would flourish.
In a anarcho-capitalist society the restrictions are dropped and everything is provided in a free and open market. Given that business in our current capitalist system are always pushing the limits as far as the law goes, here they would have no restrictions. I believe you would see a drastic climb in crime, but I believe it would be most impressive for those that are white collar. They would be the prime criminals in this system.
Every society needs to have some form of punishment that they can inflict on people who cross boundaries and break laws or societal norms. If there was no punishment then people will push the line further and further and no that there would be no retribution. You do need some form of punishment.