Use .find() instead of .children() like this:
- $("#file_dialog").dialog().parent('.ui-dialog').find('.ui-dialog-titlebar-close').hide();
However (assuming this is your initialization code), this will hide the X all the time, not just when dragging. If you want to hide only while dragging put this in a function as the dragStart option and then use a .show() on the dragStop.
Hth,
Dave