You would have to edit bfa_header_config.php for this and insert a new <tr><td>...</td><td>..</td></tr> line below the existing one (that contains the logo).
For your version 3.2
Replace
PHP Code:
$logo_area .= '</tr></table>';
// END of Logo Area
}
with
PHP Code:
$logo_area .= '</tr><tr><td></td>
<td valign="bottom" class="search-box" align="right">
<div class="searchbox">
<form method="get" class="searchform" action="' .
get_bloginfo( 'url' ) . '/"><div class="searchbox-form">
<input type="text" class="text inputblur" onfocus="this.value=\''.
(get_search_query() ? get_search_query() : '' ).'\'"
value="' . (get_search_query() ? get_search_query() :
$bfa_ata_searchbox_text ) . '" onblur="this.value=\'' .
(get_search_query() ? get_search_query() :
$bfa_ata_searchbox_text ).'\'" name="s" />
</div>
</form>
</div>
</td>
</tr></table>';
// END of Logo Area
}
And turn the search box off at Theme Options -> Header -> Show search box?