You should launch the game using "Launch Airships". Only use "Launch with System Java" if you have to.
Short Version: If you couldn't get Airships to work before, try again using the "Launch using System Java" option.
This new version should allow more people to run the game using System Java. Previously, the game would crash if launched using System Java and there was a 64-bit version of Java installed on your machine.
Now, it runs correctly, albeit with Steam integration disabled. This makes little difference in practice: your Airships account and Steam accounts are no longer synced, and your achievements are stored locally rather than on Steam.
If you can't run Airships using the standard launch option and really want Steam integration, you can get it to work by installing a 32-bit version of Java if necessary and then uninstalling any 64-bit versions you have.
Of course, this is a bit of an interim solution, and I will continue to explore options to get Steam integration working for everyone.
Also, note that the preferred way to launch Airships is just using the standard launch option. System Java is a fallback!