// Remove default header opening markup function
remove_action( 'genesis_header', 'genesis_header_markup_open', 5 );

// Add back header opening markup function w/o the structural wrap
add_action( 'genesis_header', 'sk_header_markup_open', 5 );
function sk_header_markup_open() {

	genesis_markup( array(
		'html5'   => '<header %s>',
		'xhtml'   => '<div id="header">',
		'context' => 'site-header',
	) );

	// genesis_structural_wrap( 'header' );

}

// Add toolbar inside header
add_action( 'genesis_header', 'sk_toolbar', 7 );
function sk_toolbar() {

	echo "toolbar code comes here";

	// Maybe add opening .wrap div tag with header context.
	genesis_structural_wrap( 'header' );

}