Sure thing.
In the DirectCSS section for the style used for the Store you want to do this for (SYSTEM OPTIONS => Styles Setup => DIRECT CSS), you can hide the elements this way.
For the search box:
#clipart_search_box { display:none; }
for the browse clipart on top, 2 ways.
If you want to add the entire block (inlcuding the text on top):
#local_table_clipspopup_mainsections { display:none; }
If you want to hide ONLY the "browse" button:
#local_table_clipspopup_mainsections_browse { display:none; }