This requires you to modify your masterpage just a bit. It's minimal.
In your masterpage, find id="s4-ribbonrow"
and add this to it:
style="display:none;"
Then add this after where that div ends
<Sharepoint:SPSecurityTrimmedControl runat="server" Permissions="FullMask">
<script type="text/javascript">
document.getElementById("s4-ribbonrow").style.display = "block";
</script>
</Sharepoint:SPSecurityTrimmedControl>
Your result should be something like this:
<div id="s4-ribbonrow" class="s4-pr s4-ribbonrowhidetitle" style="display:none;">
...
</div>
<Sharepoint:SPSecurityTrimmedControl runat="server" Permissions="FullMask">
<script type="text/javascript">
document.getElementById("s4-ribbonrow").style.display = "block";
</script>
</Sharepoint:SPSecurityTrimmedControl>
This will only allow users with Full Control to see the Ribbon.
Read more about the Permissions to choose between here