How to stop notification banners on a Mac (without missing anything)

Per app

System Settings → Notifications, choose the app, and pick its alert style:

• None: no banner; notifications still collect in Notification Centre.
• A banner that slides away by itself (Temporary, or Banners in older macOS).
• An alert that stays until you deal with it (Persistent, or Alerts in older macOS).

Turn off Allow notifications to silence an app completely.

For a while: Focus

Open Control Centre → Focus and pick Do Not Disturb, or a Focus you set up in System Settings → Focus, to hold banners for a while. Allowed people and apps still get through.

Hide what’s in them

System Settings → Notifications → Show previews: When Unlocked or Never keeps message text off the screen when others can see it.

Move them into the notch

NotchMind, which I make, can show notifications from any app in the MacBook notch and hide the corner banners, so you see each one where you’re already looking. Messages and Mail get the sender’s name and photo. See notifications in the notch.