You can hide from the public and registered members your postings and hide your resume download link as well.
From the "My Postings" link.
Click on step 4, Review, Edit, Delete.
From the Selection page scroll down until you see.
Show or Hide Work Experience from View
Select the post you would like to hide and then click the Show / Hide button.
FYI: If you hide all of your posts, your resume link will not display.