I think I am beginning to understand you question. The problem many (most?) answer have had is that basic income would not remove such crimes since people would still run into situations where the income would be insufficient to cover their expenses. Poverty in this context is not about income or even wealth, it is about not having enough cash: it's a cash flow problem such as one that some banks or even countries have suffered in recent years.
Such problem can't be removed by a static basic income not even one that tries to compensate for known factors. People would still have problems. Errare humanum est and so on.
In addition to basic income you would still need active intervention to deal with transient problems and special circumstances. There would have to be some instance where people could turn to when in need of help instead of turning to crime.
In modern welfare systems that is generally either the municipality and its social services as in the Nordic countries and presumably rest of Europe or a private enterprise offering short term loans. Typically for a problem like gambling or (other) addiction you would first try to cover it with loans and then after running out of credit turn to the social services. Unfortunately the social services can't really force you to get rid of your addiction or other problem. The best they can do is to make their help conditional on personal effort. Which for people with serious problems is useless and leaves crime as the last resort. Or first if the person already knows the system can't help him. Or just thinks that crime is easier than dealing with the system.
So what is needed is a system that pro-actively watches for such issues without relying on people self reporting and then enforces treatment or therapy to fix them. With that addition on top of basic income and normal social services we already have, I believe you really could remove such crimes as a meaningful factor.
My naming sense totally sucks, but it would amuse me call this "big brother" service "the Police" for historical reasons. I am something of a history buff and all.
So now that we have dealt with the preliminaries of how such society could exist, we can move to the actual question: "What would the police mostly do in such society?" We also know the answer: they would be actively preventing crime.
The system needed for this kind of observation would probably be highly automated as it wouldn't really be designed to deal with people actively trying to avoid detection. Just combine and analyse some economic, medical, and social data and flag "bad patterns" for direct confirmation by the police force or social services. So it probably would be less intrusive than Facebook or Google despite being even more of a constant presence.
The same system can be extended to many other forms of crime that have regular patterns preceding them. A society like this totally would build psychological profiles of its citizens and then intervene before the issue generates into something that is expensive or even impossible to treat.
This would leave crimes motivated by politics or religion. Honestly most crimes considered as such are caused by mental health issues and would be prevented as normal.
So I am guessing the police would mostly be doing interventions by visiting people flagged by the system with a social worker or maintaining order in areas with intoxicated people or potentially unstable foreigners. And of course just being visibly present. Although a policy of pro-active intervention would make up for lot of presence.
I doubt this is the answer you wanted or the society you imagined, but it is an answer to your question, at least.
And there would still be crime so you would still need some people to investigate such.