Hiding the Site Actions button on Public Facing SharePoint Foundation (or Server) sites:

• Open up SharePoint Designer
• Open the Master Page being utilized for your sites
• Locate the following snippet of code:
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  <span class="ms-siteactionsmenu" id="siteactiontd">
• Add:
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<SharePoint:SPSecurityTrimmedControl runat="server" Permissions="ManageSubWebs">just before the line of code in 3.
• Right click on the tag in step 3 and click on Select Tag.
• Enter this after the span tag: </SharePoint:SPSecurityTrimmedControl>.
• Save the Master Page.
• Approve and publish if required (server version).