The answer will be different depending on the country your webserver is in. It may also vary by the country of residence of you or a person mentioned, and by the country the site is being accessed from. So, in short, there are no easy answers!
The European Union, Canada, and Mexico (among others) have privacy laws that protect the personal information (such as birth date) of living people - the US is way behind the rest of the world for this.
The general rule in most countries is simply don't publish information that would identify a living person without their permission (there may be fines for ignoring this). Some countries require that permission to be given in advance in writing, in a specific form.
Even storing (rather than publishing) personal information requires registration as a "data controller" in some countries, but there's normally an exemption for "personal" use (when storing, not for publishing).
Password protecting helps, but just look at the "private" photos on sites like Ancestry to see what happens - somebody shares their "private" data with a relative, and that relative then publishes it online without any privacy at all.
Even with a passworded site, you're releasing the data to a group of people who may not have the slightest clue they should keep it private.
Personally I would not publish any information on a living person that would identify them - so maybe "private" and a birth year, to show somebody exists, but nothing more.