Twig conditional render if specific field in array of referenced entities contains text
I'd always do data checks on the entity object (in this case node), and not on the render array content. The second part is the magic .entity to get the linked referenced entity from the host's entity reference field:
{% set render_something = false %}
{% for field_item in node.field_entity_reference %}
  {% if not field_item.entity.field_to_check.isEmpty %}
    {% set render_something = true %}
  {% endif %}
{% endfor %}

{% if render_something %}
  &lt;div class=&quot;my_wrapper&quot;&gt;
    ...content...
  &lt;/div&gt;
{% endif %}

Be warned though data behind .entity is usually not cache-safe (but it is cache-safe in this special case with Paragraphs, because their revisioning is directly tied to their host entity)