From a philosophical perspective, punishment for crimes is mainly used as a deterrent to further crimes. You don't cut off a thief's hands because they've been a bad boy, you do it so they don't steal again and so other people will be too afraid to steal at all. Now, whether such punishment actually lowers crime rates is debatable (not the hand-cutting part, I hear that works wonders, but things like prison), but I would say if we lived in a world where we knew who was going to do what, and when, punishment would be replaced by prevention.
Let's keep going with the hand-cutting analogy. Say someone is going to steal; instead of cutting his hands off after the fact, just put him in handcuffs before he does anything. Since you know what crimes are going to happen in the future, you just detain him until you know he's not going to steal any more, or until he appeals for release (because of 'minority reports', you're going to want to give the wrongfully accused a chance to go free). For more heinous crimes, you might have to keep the suspect detained for longer periods of time before their futures become bright enough to set them free (now this is beginning to remind me of the anime Psycho-pass). Perhaps the government can provide counseling for these suspects (remember, you're innocent until proven guilty, and you can't prove someone guilty when they haven't committed the crime), in the hopes that they will think better of their criminal plans faster.
I'd think that many of these cases will look like imprisonment, but won't feel like imprisonment. If the government knows about and can stop all crimes, no one will ever have the chance to become a hardened criminal; they'll simply get arrested at a young age, then released once they've learned the error of their ways. The only problem you run into is when people need to commit crimes, such as poor people who need to steal in order to survive. In these cases, though, if the people are in jail, getting food and shelter, and not surrounded by terrifying people, then maybe it'll actually be a pretty good solution to poverty.
The main problem I see with this is the public's reaction. If you can get this system up and running, people may end up being fine with it (oh, you think I'm too stressed out and about to kill my boss? You're going to take me away from my job for a few weeks, so I can rest up and reevaluate my life? Sounds great!), but the biggest problem is getting it started. There will be a lot of cases that appear to not make sense, where innocent people are incarcerated for seemingly no reason. Plus, there will still be really bad guys in the prisons, so it's possible what started as a preventative measure will turn decent people into even worse criminals. And since you know what people are going to do, someone you put in jail for thinking about stealing might end up staying in jail their entire life, as each day they think of a new way to kill you for holding them so long.
There's also the problem of giving the government so much power. If the technology behind it is sound and the people using it are very transparent about their methods, it might be okay, but people generally don't like it when the government knows things about them. Maybe if there is another large terrorist threat, people might give up some freedom in exchange for safety, but due to the problems I've already mentioned I don't think they would get used to the changes before deciding to abolish them.
So, long story short, the end result sounds pretty good to me, but I don't think it could ever get started without pretty much everyone opposing it.
EDIT: I would also like to point out the case of car insurance. People my age (early 20s) don't think it's fair that they get charged more just because they're more statistically likely to get into accidents. Insurance is a really great example of pre-crime-punishment in modern society, as people have to pay for things they may never actually do. And I think people's reaction to insurance is a pretty good indicator that this kind of crime prevention system wouldn't really work.