There is no way to turn off only the sink tone but still hear audible climb indications. There is also no way to adjust the volume of just that gadget.
You can toggle all audio from the gadget by clicking on the round speaker button in the lower right corner.
You can adjust the volume of all "effect" sounds, which includes but is not limited to the variometer, in the Settings dialog. The setting for that is located here: Audio -> Application Sounds -> Effects Voume.