I was aware of this bug, I just haven't had time to patch it. Thanks for the reminder. Run an update and that should fix all of those issues. If it persists, replicate the issue on the demo site http://eventcalendar.bluehousegroup.com so I can take a look.
Ahh - After restartinig my Safari process the datepicker dropdown does not work here either.
Actually the same thing also happened for me in version event_calendar-r78 but after a total ss reinstall and upgrade to event_calendar-r79 it started working again. So maybe it's not a question of browsers but it looks like that something breaks after a while of use??
Man this drove me crazy! The instructions should be update to show that, the dataobject manager module must be installed before the calender widget will work in event clender. After I installed the dataobject manager, worked like a charm on IE8 and FireFox.