A lot compared to other countries, but not so many that a tourist would have to worry about it, especially if they stick to touristey areas.

If someone decided to go into certain neighbourhoods in certain cities, especially while looking like a tourist, they could get in trouble. But, not visiting the US because you're afraid of getting shot is like not being willing to swim anywhere in the Atlantic ocean because you're afraid of being bitten by a shark. In both cases, the danger is minimal unless you ignore the warning signs.