For logged-in users, you could do this by setting a default value for the preference which controls these popups:

$wgDefaultUserOptions['visualeditor-hideusered'] = 1;

I just looked at our code and apparently we ignore the preference entirely for non-logged-in users, which seems like a mistake. (Non-logged-in users can't set their preferences, but the default values still affect them.) I'll write a patch for that.

Change 545366 had a related patch set uploaded (by Bartosz Dziewoński; owner: Bartosz Dziewoński):
[mediawiki/extensions/VisualEditor@master] ve.ui.MWEducationPopupTool: Respect default preferences for anon users

https://gerrit.wikimedia.org/r/545366

@matmarex: Does this apply to all help messages? Not just the beta welcome

It applies to the "user education popups" in the toolbar, indicated by the blue pulsating dots.

The beta welcome dialog has its own proper setting, $wgVisualEditorShowBetaWelcome.

User education popups	Beta welcome dialog
$wgDefaultUserOptions['visualeditor-hideusered'] = 1 / 0	$wgVisualEditorShowBetaWelcome = false / true

Change 545366 merged by jenkins-bot:
[mediawiki/extensions/VisualEditor@master] ve.ui.MWEducationPopupTool: Respect default preferences for anon users

https://gerrit.wikimedia.org/r/545366