Silver Falls state park is FAR from the worst when it comes to narrow trails that hang out over the edge of cliffs. In fact, many trails in Oregon are more hazardous. However, inexperienced walkers or hikers are also far more likely to come to this park due to its fame and general popularity among the general population over some of the other places where the exact same, or more difficult, conditions exist.
Keep track of your younger children, so that they don't fall over the edge. If it has been cold, watch for ice - even if you don't think it has been below freezing, it may very well have been in certain parts of the canyon floor due to microclimates that form there.
The trail surface is carved out of rock in places, and rough. Gravel used to pave the trail and make it smooth gets knocked away very easily due to the number of visitors walking on the paths, leaving some areas with hard rocks that are easy to trip over if you are not careful.
When you come around a narrow ledge, keep in mind you have no idea what might be around the corner. It might be some jogger with headphones or someone else who is oblivious to other people, and might collide with them if you are not somewhat prepared to meet someone.
The walkways under the falls are particularly dangerous. They are always wearing away due to the water falling on them, and therefore extra rough. They are also slippery due to the water from the waterfall.
HOWEVER as long as you use some sense about this not being a standard playground, but an actual mountanous area that, except for the number of people, is a preserved wilderness, you should be just fine here.
Be really careful about ice on the trails as well. This park is higher in the mountains than the Willamette Valley floor, so temperatures are somewhat colder here. Also, the sun seldom gets a chance to warm the bottom of the canyon, and there is certainly no shortage of water to freeze into ice. So, WATCH FOR ICE as well, as seen in photo 3.
Sometimes, though, there are those who manage to kill themselves here.
As is the case with hiking their are almost always dangers. In many places along this trail system it would be fairly easy to find a steep cliff to fall over. That would not be a fun way to spend your vacation so pay careful attention to slope conditions.
When hiking beneath the undercuts also look for any falling debris it is not common but could certainly happen. Also the trails often cut into the caverns fairly deeply and in many spots you could easily bonk your head if you're not watching for that.
Bring a water bottle. Hiking for any extended distance without water is never a good idea.
Take care and heed the signs at the top of the waterfalls. People fall off every so often regardless of the warnings here. Atop the South Falls, a father and child just went over the top not too long ago. After heavy rains, the creeks can gain a lot of momentum. Also, in the winter and early spring, the trail in the canyon can be frosted or due to erosion, can be closed - as the section between the Upper South Falls and Lower South Falls was on this particular day.