It seems like this is an incompatible change in High Sierra and not specific to yEd (e.g. see https://lists.apple.com/archives/java-dev/2017/Sep/msg00018.html). At this time, it is still unclear whether this has to be fixed by Apple or by Oracle in Java.
If you are not afraid of manually editing system files, you may try editing the yEd Info.plist file as mentioned in https://lists.apple.com/archives/java-dev/2017/Sep/msg00025.html.
Another workaround would be not installing yEd, but downloading the zipped yEd jar from the yEd download page and run yEd directly from the jar file. This requires a system-wide Java installation, though.