Am I correct in assuming that you are his supervisor?
There is no nice way. He is damaging your whole department and the work they put out.
You take him in your office or somewhere private and tell him to stop.
Most companies or insurances have resources. You could go about it this way, "Jim, it seems you are having a rough time. We do have resources that can help you deal with these issues." And outline those resources that your companies' insurance provides. If he balks, "Jim, your attitude is affecting this whole office and their work ethic. I cannot have that. It is ok to gripe once in a while, "hey, we all do it", but not on a constant, daily basis".
You can also nip him in the bud so to speak, when you hear him complaining, somehow walk by and casually, jokingly mention, "Oh, I am so sorry Jim you feel this way again today! Cheer up! Hey, is that report done?" Bean dip. A lot. Each and every time. I would hope that he would soon get the hint to stop. Then, if this does not work, do the above.
You can also talk to the other employees about ways to deflect Jim. (or whatever his name is). If Jim starts in, "Oh, sorry to hear that. Well, gotta get back to work. Talk to you later!". Don't let him finish. Don't let him hardly start to complain.