Your date of birth is already protected. Only those with need to know have access. Other information can be marked private on the directory web-app:
https://directory.churchofjesuschrist.org/
For household and individual information:
1. Select your household.
2. For individual information, select the individual.
3. Click on the blue disc with a pencil icon.
4. Click on the 'Privacy tab'.
5. For each item, select the desired level of visibility: private, ward visibility, stake visibility.
Your ward clerk should be able to help with any other questions more directly than the forum can.