This question has been answered by mick2470. See the first response.
&browserBasePath=`/home/mysite/public_html`
[[!npElFinderConnector? &disableCommands=`archive, download, cut, copy, paste, duplicate, edit, open, mkdir, mkfile, netmount, netunmount, rm, rename, quicklook, view` &browserStartPath=`/subfolder` &browserStartURL=`/subfolder` &browserBasePath=`/home/mysite/public_html/subfolder` &browserBaseURL=`/home/mysite/public_html/subfolder`]]
&browserBasePath=`/home/mysite/public_html/subfolder/`//add final slash &browserBaseURL=`/subfolder/` //make this a proper web-accessible url
Try to remove anything StartPath
&browserBasePath=`/home/mysite/public_html/subfolder/`//add final slash &browserBaseURL=`/subfolder/` //make this a proper web-accessible url