The DIV tag for the "Login" area directly above the Search box had a height of 75. When not logged in this DIV tag overlapped only the "Search:" text. When logged in that DIV tag extends further to the right covering the text box itself.
Now, I didn't skin this site originally, so I won't take credit for the inappropriate height of the DIV tag. I will, however, take credit for not thinking to check this when trying to find out what the problem was.
Feel free to delete this thread since it in no way relates to NoScript.
