I had the same issue as others here for CCleaner not opening on High Sierra (This client's machine is currently running 10.13.4)
My issue was resolved by a weird way, but hopefully it can help someone else who has this issue.
NOTE: THIS WAS DONE USING VERSION 1.15.507 (LATEST AS OF THIS POST)
Go to your Applications folder
Click on CCleaner once, hold control, then click again.
Click the option "Show Package Contents" from the list that pops up. - Once this is done, a new window will open with the contents of the CCleaner application.
Click through this hierarchy of folders in this order: "Show Package Contents">"Contents">"MacOS"
Double click on "CCleanerHelper" and accept the message that shows up. It's either "Okay" or "run" but I am not certain at this time.
After that is done, you'll see the CCleaner icon at the top of your screen.
Now double click the "CCleaner" icon to the left of "CCleanerHelper" - This will open up a terminal windows and open CCleaner with info about the launch.
CCleaner launched for me at this point. So if this helps you, leave a comment letting me know if this helps. If it doesn't, i'll see if I can invest in a machine and dive deeper into the issue!
EDIT: 1.12 showed to be too low of a version and not supporting the MacOS version 10.13.4. So please be aware of that and use this method with the latest version of CCleaner