This handy piece of JavaScript will disable the drag and drop function for all library web parts on a single page. Other upload options will still work.
Just paste this into a Script Editor web part:
<script>
ExecuteOrDelayUntilScriptLoaded(function() {
g_uploadType = DragDropMode.NOTSUPPORTED;
SPDragDropManager.DragDropMode = DragDropMode.NOTSUPPORTED;
}, "DragDrop.js");
</script>
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