You can use this setting:

https://crosstec.org/en/support/online-documentation/breezingforms/18-email-configuration/68-admin-emails.html

Report "non-empty values" which won't show the empty ones.

If you are using a custom template or something then you might prefer to use PHP.

Here is how you set up the PHP https://crosstec.org/en/support/online-documentation/breezingforms/10-advanced-settings/739-how-do-email-templates-work.html

<?php
defined('_JEXEC') or die('Direct Access to this location is not allowed.');
?>
 
<?php if ($RECORD_ID != ''): ?>
<?php echo $PROCESS_RECORDSAVEDID?> <?php echo $RECORD_ID ?><?php echo $NL ?>
<?php endif; ?>
 
<?php echo $PROCESS_FORMID?>: <?php echo $FORM ?><?php echo $NL ?>
<?php echo $PROCESS_FORMTITLE ?>: <?php echo $TITLE ?><?php echo $NL ?>
<?php echo $PROCESS_FORMNAME ?>: <?php echo $NAME ?><?php echo $NL ?>
<?php echo $PROCESS_SUBMITTEDAT ?>: <?php echo $SUBMITTED ?><?php echo $NL ?>
<?php echo $PROCESS_SUBMITTERIP ?>: <?php echo $IP ?><?php echo $NL ?>
<?php echo $PROCESS_SUBMITTERID ?>: <?php echo $SUBMITTERID ?><?php echo $NL ?>
<?php echo $PROCESS_SUBMITTERUSERNAME ?>: <?php echo $SUBMITTERUSERNAME ?><?php echo $NL ?>
<?php echo $PROCESS_SUBMITTERFULLNAME ?>: <?php echo $SUBMITTERFULLNAME ?><?php echo $NL ?>
<?php echo $PROCESS_PROVIDER ?>: <?php echo $PROVIDER ?><?php echo $NL ?>
<?php echo $PROCESS_BROWSER ?>: <?php echo $BROWSER ?><?php echo $NL ?>
<?php echo $PROCESS_OPSYS ?>: <?php echo $OPSYS ?><?php echo $NL ?>
 
<?php foreach ($MAILDATA as $DATA): ?>
<?php echo $DATA[_FF_DATA_TITLE]?>: <?php echo $DATA[_FF_DATA_VALUE]?><?php echo $NL ?>
<?php endforeach; ?> 

You could just do an if statement before the second last line.

<?php
defined('_JEXEC') or die('Direct Access to this location is not allowed.');
?>
 
<?php if ($RECORD_ID != ''): ?>
<?php echo $PROCESS_RECORDSAVEDID?> <?php echo $RECORD_ID ?><?php echo $NL ?>
<?php endif; ?>
 
<?php echo $PROCESS_FORMID?>: <?php echo $FORM ?><?php echo $NL ?>
<?php echo $PROCESS_FORMTITLE ?>: <?php echo $TITLE ?><?php echo $NL ?>
<?php echo $PROCESS_FORMNAME ?>: <?php echo $NAME ?><?php echo $NL ?>
<?php echo $PROCESS_SUBMITTEDAT ?>: <?php echo $SUBMITTED ?><?php echo $NL ?>
<?php echo $PROCESS_SUBMITTERIP ?>: <?php echo $IP ?><?php echo $NL ?>
<?php echo $PROCESS_SUBMITTERID ?>: <?php echo $SUBMITTERID ?><?php echo $NL ?>
<?php echo $PROCESS_SUBMITTERUSERNAME ?>: <?php echo $SUBMITTERUSERNAME ?><?php echo $NL ?>
<?php echo $PROCESS_SUBMITTERFULLNAME ?>: <?php echo $SUBMITTERFULLNAME ?><?php echo $NL ?>
<?php echo $PROCESS_PROVIDER ?>: <?php echo $PROVIDER ?><?php echo $NL ?>
<?php echo $PROCESS_BROWSER ?>: <?php echo $BROWSER ?><?php echo $NL ?>
<?php echo $PROCESS_OPSYS ?>: <?php echo $OPSYS ?><?php echo $NL ?>
 
<?php foreach ($MAILDATA as $DATA): ?>
<?php if($DATA[_FF_DATA_VALUE] !== "") { ?>
<?php echo $DATA[_FF_DATA_TITLE]?>: <?php echo $DATA[_FF_DATA_VALUE]?><?php echo $NL ?>
<?php } ?>
<?php endforeach; ?> 

You might want to do some error checking in case you need null instead of "" or empty or something instead.