[verb ない-form] + でください [verb ない-form] + で
~ないでください asks someone politely not to do something: "please don't". Take the ない-form of the verb and add でください.
The で is part of the pattern, and it follows ない, not なく.
Drop ください and ~ないで on its own is the casual request. It is still a request, softer than the blunt な of すう吸うな.