Hiding Sign In Link For Anonymous Access Users in SharePoint

To customize SharePoint “Sign In” link, we need to modify or duplicate Welcome.ascx in your Control Templates directory.

Here’s a good post specifying how to hide your sign in link.

Hide the Sign In link for the anonymous access user in anonymous access enabled site - Bend the Welcome.ascx - SharePoint MOSS

You can play around with Welcome.ascx and come up with your custom style sign in.