Addis is safer than most cities in Africa. Gang violence and similar serious activities are unusual. However, you may encounter some pick-pockets and con-artists around and inside Bole Airport, Mercato, Piazza and many other areas. Keep your belongings close, and pay attention to your surroundings.
In a total difference of other African cities, in Addis-Ababa, police officers never approach foreigners to ask them to present a passport, ID or "legal" papers. Once you show your passport at the airport, you are free to move around pretty much anywhere. The only time you need your passport or ID is for hotel registration (booking) and other similar and few instances.