It would have almost no effect on society or the tendency to abide by such a contract i'm afraid.
Contracts mean little by themselves. They are only relevant in that there are ways to enforce them provided by the government. However, the government already does everything it can to enforce the social contract, that's what all it's laws and police officers do. Since the enforcement wouldn't change the contract wouldn't.
Look at the abstinence pledges that have become common lately. Studies show that those who sign it are more likely to end up pregnant teenagers. part of this is that the sort who are likely to sign it happen to be slightly more likely to be social outgoing individuals who are likely to engage in sex, but the main point is that the contract signature itself had very little, if any, effect on their abiding by the stated rules.
A contract that everyone was forced to sign, either officially or implicitly via social pressuring, would have even less power. Those who signed it did not do it of their own free will, and thus may not feel bound by the intent of it.
That isn't to say it couldn't have some interesting world building aspects. The signing of the contract could be something like your bar mitvah, a right of passage for young children into adulthood. That would give it a very relevant cultural impact, but the impact is less about the contract itself and more about having something to symbolize transition to adulthood. Any number of other activities could be made up with the exact same relevance.
Taking the Bar Mitvah analogy even further, before having one a boy would need to read the Torah. If those signing the contract were forced to read and prove an understanding of it that could have some relevance. The greater awareness of the contract and the fact that people can quote specific relevant passage from the contract years later may give people a better means to articulate their intent and belief when discussing legal matters. Discussions of right and wrong would likely reference the contract much the way we reference the constitution when discussing the validity of rules today.
However, I think in general the net crime and abuse will be the same. Human nature is the same regardless of the culture we place it in. The way one articulates rights and wrongs may change, but in the end people will be just as prone to committing wrongs, or rights I suppose.