And actively encouraging these 'weak and stupid people' to gamble their money away costs the taxpayer even more in the long run.
Scenario 1: We protect 'weak and stupid' people from themselves to a reasonable degree. Some of them might invest in education, etc. and so make more money for the government later. They'll save more too for retirement. They'll invest and that will pay money in taxes too.
Scenario 2. We suck a 'weak and stupid person dry in his early twenties by getting him hooked on casinos. He has no money for his further education, saves no money, and is more likely to suffer mental health problems like attempted suicide, loss of job, etc. We might collect some money from him in the short term, but he just sucks off the public tit for the rest of his life after that.
Now that's what we call short-term investment and long-term cost. And who pays for it? The taxpayer of course.
We should remember too that gambling addiction can push the gambler into theft, drug trafficking, or prostitution to feed the addiction. This in turn imposes costs on the criminal justice system and health care in the form of HIV/AIDS and other STI treatment. The costs of gambling addiction on the economy reach much farther than many realize.