I just accidentally found an answer to this, when I was just interested in another plugin and read its source code.
(Edit: While the following code may work, please find a better solution from my newer post further down this thread.)
//@ts-ignore
this.app.vault.adapter.basePath
In order to use basePath, you need to instruct typescript not to yell about basePath not being defined during compiling. Hence this has the line //@ts-ignore. I don’t know why this supression is needed - I mean: why the property is not defined in the first place? If anyone else knows a better way to get the path, please let me know.
The code was copied from here: obsidian-open-with/main.ts at 84f0e25ba8e8355ff83b22f4050adde4cc6763ea · phibr0/obsidian-open-with · GitHub
Thanks @phibr0 and sorry for “stealing” a couple of lines of your code! 