AirPort Utility 5.6.1 for Mavericks resolves a problem as old as OS X 10.9 itself: a system file upon which AirPort Utility relies was updated in 10.9 and that update broke backwards compatibility with AirPort Utility v5.6.1. The file in question is Apple80211 in /System/Library/PrivateFrameworks/Apple80211.framework/Versions/A. Replacing this file with the older version allows AirPort Utility v5.6.1 to be run under 10.9, but only via Terminal (i.e., double-clicking it will not launch it, it will throw an error).
However, there is a way to get around this issue by importing the older library into AirPort Utility v5.6.1 before running it. That is what this new solution does.
Because the application is not code-signed, you will need to follow this procedure to run it the first time. After that, it should open without complaint.