Check Safari > Preferences > Advanced
Click None Selected from the Style sheet pop up menu.
And check Safari > Preferences > Extensions
If there are any installed, turn that OFF, quit and relaunch Safari to test.
If it's not an extension issue, deselect Enable plug ins in Preferences > Security
Quit and relaunch Safari. If that helped, then there's one or more third party plugins causing problems with your drop down menus.
Troubleshooting help here > Unsupported third-party add-ons may cause Safari to unexpectedly quit or have performance issues