I never would have said this a year ago, but sometimes giving negative stakeholders some attention and TLC can convert them into powerful allies for your project. In your example, if their job is being eliminated, there's nothing you can do. However, in one of my recent projects, we took a negative stakeholder aside, showed him how even though the project seemed like a headache for him, it was good for the company -- and he liked being employed by the company! In another instance, we gave a negative stakeholder an opportunity to give input into the product design. We haven't actually implemented her idea, but feeling like she had contributed and had ownership changed her attitude tremendously.
Converting negatives to positives is probably the exception rather than the norm, and I would not spend a ton of time and energy trying to do it. However, if you are able to succeed, it can reap great benefits. In both of my examples, the converts turned into great project champions.