Use Genesis Attributes to add Inline Styles
<?php
// Add background-color to .footer-widgets
add_filter( 'genesis_attr_footer-widgets', 'ap_attributes_footer_widgets' );
function ap_attributes_footer_widgets( $attributes ) {
$background = get_field('footer_widgets_bg', option);
$attributes['style'] = 'background-color:' . $background . ';';
return $attributes;
}