You can add classes in the twig template directly, or through a preprocess function.
Preprocess
Docs provide some patterns to follow: https://api.drupal.org/api/drupal/core%21includes%21theme.inc/function/template_preprocess_field/8.2.x
function HOOK_preprocess_field(&$variables) {
$element = $variables['element'];
switch ($element['#field_name']) {
case 'MY_FIELD_NAME':
$delta = 0;
while(!empty($element[$delta])) {
$variables['items'][$delta]['attributes']->setAttribute('class', 'custom-class-name');
$delta++;
}
break;
}
}
This applied the class to the entire entity, alongside field__item.
Template
If you have twig debugging enabled, you can see quickly what's a good template name. I tested with field--node--FIELD-NAME.html.twig. Using the field template from classy for example, just add the class at the end of the list.
{%
set classes = [
'field',
'field--name-' ~ field_name|clean_class,
'field--type-' ~ field_type|clean_class,
'field--label-' ~ label_display,
'custom-class-name',
]
%}
This applied the class just to the field, not the label. But you could modify the template to print the classes elsewhere.