Make a list of things you need to keep an eye on. Start with the obvious things, like the sidewalks, railings, parking lots, etc. As you think of things, add them in. Keeping a growing list is one of the best ideas when it comes to organization. How are the lights around the complex? Do you have a laundry room, and if so is it clean, bright, and safe? Do you have a plan for winter with the snow and ice? Continue like this until you cannot think of any more.
Ask around the complex itself. See what your tenants are concerned about. They may come up with things you hadn't thought of, like dead trees or the need for an extra dumpster or recycling service in the back. Perhaps they saw a growth of poison ivy by the back fence that you had no idea about. It's always a good idea to check with those who live at the complex, since they spend most of their time looking at the area.
Keeping a list of things to check is the easiest way to ensure your complex's safety. See about a couple of things each week and if they're good, mark them off on your list until the next time. This can help keep your anxiety down and your productivity up. It's a win for everyone this way!