Detroit had an industrial based economy. When the jobs left, people didn’t simply decide to go on welfare. The people left as well to look for employment elsewhere. The population of Detroit has declined from about 1.5 million in the 1970s (when manufacturing jobs started to leave) to a population of about 700k now. Detroit has lost about 25% of its population in the last 10 years.
With the loss of jobs, businesses, and population, came the loss of city revenues needed to run and support the city, so city services began to disappear, leading to further decline. Unfortunately, Detroit has not been able to attract enough new business to replace what it lost.
Like I said, there were many contributing factors to why this all happened, but welfare programs really wasn’t one of them.