Could be an SR13 issue. We had a number of issues with moving over from SR6 to SR13.
Maybe check the attached release notes for SR14 for the issues resolved from SR13 and previous versions.
Have you tried rebooting Qv services and restarting your browser or creating a custom object on AJAX and adding dropdown select?
This is an old post but I just came across the same issue in QV 12.20.20400.0. In QlikView desktop the drop down select arrows were present, on the AccessPoint they were no where to be seen.
I was able to fix it by changing the settings on the straight table for Sort Indicator, Selection Indicators and the Null and Missing symbols to as per below.
Just in case anyone else finds their way here.