core.dmp files hogging space on C:\

these are system dmp files which are triggered on an event of running Native or Heap out of memory , depending on what you have set as -Xdump JVM argument

These files in huge in size , depending on how much heap is configured and how much was used when the process Ran out of memory,

I would expect your server not to work if that happens.

Otherwise check Xdump argument to see if has argument to write system dump when ever you take a JAVA core.

you can delete them, its safe to do it

Depending on their age (as you say, the file names contain timestamps), if they're very old then you can probably just delete them as too much time has likely passed for them to be of any value.  If however they're recent, it's an indication that your WebSphere server is running into problems and is crashing so you would do well to move them somewhere safe and begin an investigation into the issue that's causing the crashes.

They are often accompanied by javacores, heapdumps, snapdumps etc. All these files can be used to diagnose problems within the WebSphere server so if you do move them you should keep them together.