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 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.