Unfortunately no. Shifter has no password protection feature for generated page or posts also has no Basic Auth feature.
We only support non-password protected page or posts, of course amp posts/pages too.
When Shifter generates password protected page, it will display the content as "This content is password protected."
If you're paid plan user, it's a good idea to get an artifact then upload to other hosting services (e.g. Netlify or GoDaddy) and use their Basic Auth feature.