In the CuteEditor_Files/Scripts folder, edit "insert_table.aspx" and change line 16 which is a javascript code declaring var editdoc and find "border=1" in that string and change it to "border=0".
Line 16:
Scroll over and find this:
And change it to:
<!-- Inject Script Filtered -->
<!-- Inject Script Filtered -->
<!-- Inject Script Filtered -->