Compared with the other major cities of the world, Tokyo is a very safe city with a low crime rate. This is one of the few areas in the world when you could be walking on any street safely at three in the morning.
However, do exercise caution when visiting the Kabukicho district (east of Shinjuku station) at night. If someone approaches you with cheap offers for drinks, you can be 120% sure that you will be ripped off. When the hawkish Governor Ishihara was at the helm, the city did a nice job of getting the touts off the streets. Under the current Governor, unfortunately the reigns seem to have loosened and the guys are back bugging locals and tourists alike. Simply ignore them and you will be ok.
The Japanese have a rather high tolerance for being drunk in public. If you see someone acting stupid at night, just leave them alone.
Earthquakes occur on a regular basis which is why the buildings in Japan must adhere to many strict regulations. By now most buildings have been retrofitted or built after 1981 when the requirements got tougher. The biggest concern will be of tidal waves. If you are in a coastal area when you encounter reasonably strong shaking, it's always good to take a precautionary measure by moving to higher ground immediately.
Watch for those bicycles!
It sounds strange, but the biggest danger for visitors to Japan might be the risk of being hit by a bicycle speeding along on sidewalks. Traffic laws were amended back in 2015 prohibiting bicycles on sidewalks unless specifically mentioned on road signs. Unfortunately the rules are hardly ever observed, or in some cases the roads are too narrow to make it realistically impossible for bikes to run on the roadside. We have encountered aggressive behavior by some cyclists which has alarmed some of our participants during tours. If you're exiting from a building or at an intersection, keep an eye not on just vehicles but also bicycles as well!