How to diplay a div only if at least one author custom profile field is filled?

One possible solution could be this –

if ( empty( $social_icons_fields ) ){
   $icons .= '<style type="text/css"> .socials-icons{display: none;} </style>'; //assuming that the div which has class 'socials-icons' is the div which supposed to be not shown 
   return $icons;
}