In this case, if the Block was created from a view (if not then you can create one) you can set the access settings in the view edit mode:
By selecting permissions you can make access to this block fine grained, based on rarely granted permissions such as 'administer-permissions' which is a permission only Administrators are likely to have. click on the little 'cog-wheel' to the right of the 'Access' row under the view's basic settings in order to select the restriction you desire:
Please pay careful attention to the comment below the 'Default Access Options' list, as shown in the third image. It reads 'Only users with the selected permission flag will be able to access this display. Note that users with "access all views" can see any view, regardless of other permissions'.
You can remove the 'access all views' for the anonymous user under '/admin/user/permissions' and then set the permissions of your other views also. This gives more control over who sees what. Views by default have 'Access' set to 'Unrestricted' generally.