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It's very difficult. This problem has nothing to do with the workplace, it is embedded in the some people think.
The solution is to have a relentless positive attitude, and insist on such an attitude from your staff. Do not allow people to talk negatively: work is about achieving a sequence of goals, and any negative remark can be addressed by "Ok, but how does that remark help us achieve goal X? Because if it doesn't, please don't say it. We are being paid to achieve goal X, so that's what we talk about."
People who have fallen into a habit of negative thinking because of negative experiences will adapt. People who are determined to be negative (and there are some) will eventually leave or cause so much trouble you can fire them.
