This is a great question. I think FDR's statement is only partially true. I think that the times have changed fundamentally that there are very legitimate things to fear. Let me give you a few examples.
First, in a post 911 world, the threat of terrorism is real. In view of all the wars in our world and the sophistication of our weapons, another war is a real possibility. Moreover, war can be extremely devastating.
Second, we live in a different world economy. One or two countries are not hurting but many more. In fact, all of the advanced Western economies are hurting. Moreover, in a globalized world, we are all connected. The contagion can be great. This is something that we need to be aware of.
Third, our country seems more polarized than before. This, too, is troublesome. Political fights seem to escalate beyond the past. Remember a few Republican almost made our economy default.
In light of these points, there are legitimate fears. To neglect these points is probably to be foolhardy.