yeah, reform your country so that it not only has universal health care (incl. mental health) but also has universal unemployment benefits and disability pensions so that people who would be broke as well as homeless, unemployed and/or crazy in your current system would have a small income on which to feed, clothe, and house themselves and therefore wouldn't be so desperate they had no choice but to mug strangers for a living.
it sounds like it would be expensive but it's much cheaper than prison and much cheaper than the cost of theft-related murders, and the cost of street crime in general.
WTF do you think countries like australia or the UK or much of europe don't have rates of begging or mugging even a tiny fraction of that in the US? we have a tiny handful of real crazies doing it, you have the crazies and all the desperate poor doing it as well. you also have a gang culture because neighbourhood gangs take the place of a social safety net in a country without welfare entitlements for all.
also, you need to restrict gun ownership, especially hand-guns - start with the obvious things like serious background checks and require owners to demonstrate a *real* need to have a gun (not just "i'm scared of black/poor people").
fuck it, you also need a revolution. send all corporate fat-cats and lobbyists and corrupt politicians to the guillotine. that's about as likely as the US doing any of the other things above.