Apple Remote Desktop (ARD) 3.7 - how to drag n drop copy files from local to remote
I can confirm that the file copy still works on the newer versions including ARD 3.7.1 so it's something you'll need to use logs to troubleshoot. 

You may have to use the copy tool to try copying the file and look over the logs to determine what has changed between your remote computers and the controlling remote desktop computer.

That interface has better error messages and reporting and has helped me figure out what firewall settings were preventing me from using ARD in the past. Feel free to post your log if you can't make sense of the details it lists.