Have you tried:
– deactivating all plugins to see if this resolves the problem. If this works, re-activate the plugins one by one until you find the problematic plugin(s).
– switching to the default theme to rule out any theme-specific problems.
– resetting the plugins folder by FTP or PhpMyAdmin. Sometimes, an apparently inactive plugin can still cause problems.
– re-uploading all files & folders – except the wp-content folder and wp-config.php & root .htaccess files – from a fresh download of WordPress. Make sure that you delete the old copies of files & folder before uploading the new ones.
Incidentally, there has never been a font dropdown by default in WordPress. Perhaps you were using a plugin for this previously?