How do I turn off getting notifications about certificates for sites I know that are OK?

Ah, this has been tightened down further—the View Certificate > Details display used to have a "Trust" button, and it's no longer present in High Sierra. When I select the Visit This Website link, I get a password prompt, and when I enter that I can then re-visit the site without a prompt.

(This path is not available to iOS, too.)

Fix the site, or access the site using HTTP and not HTTPS.

This is not going to get easier.

The general path...

Show Details...

Then tried the view certificate (which is where the "trust" button used to be)...

Following is visiting the site anyway...

Password dialog that pops here is not shown.

Subsequent access to the site did work, and did not generate the certificate error sequence.

Maybe you're hitting a pool of different servers with different (bad) certificates?

Again, iOS has greater difficulty getting past a bad certificate than does macOS, and this whole bad-certificate warning is only going to get more difficult to bypass in general. Get the certificate fixed, or get rid of the certificate, or get the public root cert loaded onto all of the clients if y'all are working with a privately-issued certificate chain; with your own private CA.

Drill into the notice and mark them as trusted, depending on the macOS release.

Or work with the folks to fix the site—certs from letsencrypt or otherwise—as these mismatches are only going to get more problematic.

Or both.

You can view or change a certificate’strust policies in Keychain Access.

Open Keychain Access for me

In the Category list, select a category.

Select a certificate, then choose File > Get Info.

Click the Trust disclosure triangle to display the trust policies for the certificate.

To override the trust policies, choose new trust settings from the pop-up menus.

Safari Certificate not recognized

Certificate isn’t being accepted