You'll need to look at the attributes that you can assign to the SEARCH skin object.

Here is a blog entry about the SEARCH skin object.

Two attributes are ShowWeb and ShowSite. 

Per the blog entry, make sure that you set UseDropdownList to false, and that you set ShowWeb and ShowSite to true or false, depending on what you want.

If you are using HTML skins, you do this in the skin.xml file.  If you are using ascx skins, you do this in the skin file.

Look at the file admin\Skins\Search.ascx.vb to look at the codebehind page for the SEARCH skin object.  If you look through the code, you will see all of the attributes of the skin object.

we have covered this in a tutorial here:
How to style and skin the search box in DotNetNuke