Is there any way to mitigate the problem without entirely disabling auto-responses?
Yes, due to recent attempts to “fix” the problem of forged email, there is. If you must continue to use an auto-responder, you can greatly increase it’s accuracy (and perhaps avoid being blocked). Using this method, the auto-responder may not always respond to every legitimate email. It will respond to the vast majority, and it will send much less (although not zero) misdirected mail. To do this, your responder should use SPF and/or Domain Keys to verify the authenticity of the message being replied to.