The answer depends partly on your definition of "crime" and whether international law is suitable for dealing with criminal issues. Adopting the prevailing viewpoints, however, legal issues related to activities in space are addressed by a set of international treaties, most notably the Outer Space Treaty of 1967 (OST), which has been signed and ratified by all "space-faring" nations. Under this treaty, countries have agreed essentially to be responsible for all activities in space that are conducted by their respective "entities", including governmental and non-governmental entities and private individuals. So if a citizen of Country X causes harm to a citizen of Country Y in space, then (assuming both countries are signatories of the OST) Country X has an obligation to restore the damages to Country Y.
What this means in terms of specific "crimes" committed by specific people against other people is largely an untested question. As someone else mentioned, there are more specific agreements between countries for specific cases, such as the Intergovernmental Agreement of the ISS.
Even though the OST addresses non-governmental entities, it is largely geared towards governmental actors in space. For example, if the US launches a rocket and it accidentally falls on, e.g. the Palace of Versailles, the US would probably be shelling out a large reparation payment to the French Republic. Countries generally supervise and restrict their own citizens from launching things in space, unless they insure those activities. For example, rocket launches conducted by commercial companies are almost always insured against third-party damages.
Your question, however, speaks to a big open hole in space law. If private individuals will start to operate and even live in space, Earth-bound governments may no longer wish to remain responsible for their actions. For example, one could imagine that a country might rescind the citizenship of a "rouge" space-traveler, precisely in order to avoid liabilities caused by this "space pirate". Countries would probably still be expected to provide protection and assistance to their own space-based citizens and assets against such pirates, but prosecuting the pirates criminally (or under an international legal framework) might be near impossible.
To gain deeper insight, note that the OST treaty and much of space law is largely based on long-standing customs and treaties concerning activities in international waters (e.g. Convention on the High Seas). If you think about it, international waters are very similar to space in terms of jurisdiction. Every vessel is normally required to operate under a certain country's flag, and that country then assumes responsibilities for certain liabilities. While there are certainly some pirate vessels (even today), usually the actions of those vessels are traced back to a particular country, and international pressure is applied to that country to control the pirates' actions and perhaps in seeking reparation for damages. Many wars have erupted or escalated precisely because pirates from a particular country committed crimes against ships of another country.
The Wikipedia page for Space Law is also a good starting place for learning more about these issues.