Compared to war zones like Guatemala City or Tegucigalpa, Panama City is relatively safe. That being said, you still need to be cautious. There are a lot of neighborhoods you cannot walk through, and good areas can turn to bad ones quickly. Casco Viejo, for instance, is home to the Presidential Palace and is one of the most heavily policed areas of the city. But if you stroll west of Parque Santa Ana or north of Plaza Herrera, you'll see the neighborhood degenerate into a dangerous slum very quickly.
Taxis are generally a better idea than walking at night. If you're planning a long walk, make sure to ask around and see if it's safe first. If a locals advise you to take a taxi, listen to them.