How do you turn off the blue alert banner about signatures

The “Verify signatures when the document is opened” setting doesn’t actually control the blue banner. It only controls automatic signature validation, so unchecking it won’t necessarily remove the banner.

If you see an X on the right side of the blue bar, you can close it for that document. On some versions, clicking the signature/status icon on the left side of the bar also hides the banner and moves the status to the side toolbar.

Unfortunately, there doesn’t appear to be a preference to permanently disable the signature status bar for all signed PDFs. Adobe treats it as part of the signature validation/status information.

If you’re on a Mac and don’t have the X option, it would be helpful to know the exact Acrobat version, since the behavior can differ between the New Acrobat and older interface.

Unfortunately I don’t have the X option to close the window. That would be such an obvious thing to have. I’m on a MacBook so maybe that explains the difference. 

You are right that the 'Verify signatures when the document is opened' preference does not control the banner itself. It only controls whether Acrobat validates signatures automatically, which is why unchecking it did not remove the blue bar. That banner is the Document Message Bar showing the signature status, and Acrobat displays it for any signed document. The practical fix is to click the X at the right end of the banner, which dismisses it for that document while it stays open. Keeping automatic verification off, as you already did, plus dismissing the bar per document is the closest you can get, since there is no preference that permanently hides it for signed files.