I believe to be successful we'll need a hybrid system comprising of both rehabilitation practices and punishment practices though. The system should heavily favor rehabilitation over punishment however. Punishment should be seen as a last resort, when all other rehabilitative options have been exhausted and a citizen is still being problematic.
The criminal justice system in Asgardia in general needs to be approached from many several angles.
Prevention. Most would agree that most crimes stem from issues such as a lack of resources or mental health issues yes? So we would need to make sure that everyone has their needs met. That everyone has sufficient food, water, basic living supplies. Maybe a basic income for all adult citizens. All in equality too. We must all be living in one class. No upperclass or lower class. No preferential treatments towards any group for any reason, including members of the Asgardian government. On the other hand, we will need a comprehensive and fully modern mental health care system, one that approaches all forms of mental health issues as legitimate diseases of the human mind. Should there be existence of citizens dealing with drug/chemical addictions, that needs to be seen and treated equally as a mental health issue rather than demonizing the person and labelling them as low lives. There will come a difficult question though that needs to answered, and that's what do we do with those who have such significant mental health diseases that they cannot peacefully live within the society? Tough because it is no choice of their own, but how can we allow these people to coexist within our community knowing that there's nothing that can be done to heal or rehabilitate them? It may be necessary to have a section for such potential individuals to live in Asgardia while removed from the community. We would need to ensure their safety however and that they are provided the best possible living conditions.
Education. This one is important. We will need a very extensive and comprehensive education system that starts young. This ties into prevention as well. Educate and prevent.
Rehabilitation. For those who commit crimes, like I said, the main focus should be in rehabilitation so the person does not repeat their crimes. Find out what drove the person to said crime and find out what can be done to prevent it from happening again. Like others have mentioned, Sweden is a good example that this system works.
Punishment. It would be foolish to say that everyone and all issues can be resolved with rehabilitation. In the event that all rehabilitative efforts have been exhausted and a person is still a significant reason for concern, then we need a penal system. This could be some form of jailing system. Or perhaps just banishment from the community/colony/space station all together. If we've done everything we can to turn someone back into a productive member of society, and there's just no way they can peacefully coexist with everyone else, why should we keep them in our community and provide what is surely to be limited resources for them? If it's not an issue of severe mental health where they have no control over their actions, then banish them. Send them back to earth. I don't believe we should adopt capital punishment. Banishment should also be a consideration for those who have committed truly horrendous crimes. At the same time, the decision to banish someone can not be left to one single person. There would need to be some sort of panel of peers, akin to a jury in the US, or even a colony wide vote on the matter. With the advancement of technology, it shouldn't be difficult at all to get every to quickly and securely put a vote in.
One other aspect I think about is actual law enforcement. I don't believe law enforcement personnel should have any sort of lethal weapons. They should all be trained in de-escalation techniques, rather than forceful techniques, when in confrontational situations with alleged criminals. If that doesn't work, I saw someone mention earlier that they should be trained in martial arts and perhaps with technological advances there will be better non lethal weapon options that can be used. I think law enforcement also needs to be much more Integrated within the community. A law enforcement official should be seen as a friend, as an equal member of the society rather than an authoritative figure who should be feared.