Using both s123 3.6 and 3.7.60 I am unable to utilize the basemap switcherfor any published surveys that I have edited in Survey123 Online.
I have tried using the standard (beta) and basic map types, re-installing s123 and re-downloading the surveys. Survey123 for ArcGIS
Just to check, the workflow you are describing is:
1) Create and publish a form via the Survey123 website
2) Open the form in Survey123 Connect and proceed to the Map Settings
Is that correct? I was able to do that successfully for a test survey.