Add a script web part with the following:
<style
type="text/css">
#Ribbon\.Documents\.New\.AddDocument-Large
{
display:none
!important;
}
</style>
<style
type="text/css">
td.ms-addnew
{
display:none
!important;
}
</style>
<script
type="text/javascript">
ExecuteOrDelayUntilScriptLoaded(function()
{
g_uploadType
= DragDropMode.NOTSUPPORTED;
SPDragDropManager.DragDropMode = DragDropMode.NOTSUPPORTED;
}, "DragDrop.js");
</script>
