This is Apple's support article regarding installing the older (installer media) versions of ARD Admin on the newer OSes. The article is for Lion but does work for 10.8 and 10.9 as I've used the steps outlined many times:
An important point to remember is to run Software Update immediately after copying the application to your Applications folder. What you could do is download the update prior to extracting the Admin application. That way you'll have it ready. You may have to download previous updates first?
The other important point to remember is to leave the application where it is and not be tempted to move it.
Thanks, Antonio. That got me to the point I have ARD on the Mac Pro, but running Software Update immediately after the install was a bust. Manually downloading the 3.6.1 Admin update was also a bust. Despite the version on my Mac Pro being 3.5.1, the 3.6.1 update refuses to install by reason of the ARD Admin app being "newer". Having no other choice, I have now moved on to the 3.7.1 Admin update. It looks as though that one will install.
Closer than I was, but still a long way off.
Ok, 3.7.1 Admin Update installed. I finally got the new install to accept my old serial number and it looks like I am flying. Thank you again, Antonio.
Now if only Apple would join the 21st century and make migrating between machines easier than the PITA project it is now. No wonder I waited six years for a new computer. It's all coming back to me now.
FYI - New operating systems always require compatibility updates of Apple Remote Desktop, so it's no wonder that anything prior to ARD 3.7 would run on Mavericks. Be aware of this in the future and save yourself some headaches. Software Update should prompt you, but you can always just grab the manual retail admin updater.