Hi Mr T. I was in Osaka during the March earthquake and I swear, had I not turned on a television set I would not have known anything was amiss in the country. Your main safety concern has and always will be earthquakes - but this is a known risk when entering Japan.
Basically, anywhere around Kyobashi is fine. You'll find many cheap eateries and izakayas, and the cost of items in Japan is far more reasonable than people make it out to be (although I am from Sydney, so anywhere really to me is cheap!)
My recommendation is to take the subway and check out Osaka's centre, Dotonbori - it's lots of fun and very safe.
Jesse - World Nomads Safety Hub about 8 years ago
I stayed at an old hotel that has a public bath at Doubutsuen Mae station. It's a seedy part of town with a lot of elderly and working class. But, the hotels there are cheap and a step from the subway. I stayed at Hotel Taiyo. www.hotel-taiyo.com/en/map_e.html about 8 years ago