I'm a pilot, not a FA.
But i've learned a thing or two about passengers.
One...first approach the situation as if you knew the passenger was scared beyond belief, of what it almost doesn't matter. Be very kind...take a few minutes to talk to them (yes I know you are busy). Listen to them. Are they going to a funeral? ARe they without money? Are they ill in some way. Treat them very well, take them politely to a quiet place and talk things over. Maybe they are worried about flying and would like to meet the pilot. Maybe their luggage is lost or something else you can really help with.
OR
On occasion you have to exercise authority. I've done it once and it worked great. A passenger was very upset about going to a thanksgiving family get together and didn't want to go, she was arguing with her husband and it was a terrible situation. not a heckuva lot of time to deal with it.
After making sure the woman was not being kidnapped or anything really nutty or bad, I made it clear to them both the very nature of the world on an airliner.
I said: I want you to understand that an airliner is not a Democracy. I am the boss, PERIOD. And the flight attendants are my deputies and if you don't do what they request, I will land the plane and have you arrested. I gave them a choice of getting off the plane or living by my rules for the length of the flight.
They stayed on and there were no further problems.
First love them and care for them.
Then order them and offer them arrest or removal.
I once took a scared passenger to the cockpit during boarding...had her shake hands with the copilot and myself. Explained to her the noises and gadgets and everything I could think of her. I personally took her to her seat, helped her with her seatbelt (a mature woman, not a young hot chick) and shook her hand and told her I would be extra careful.
The flight went fine and she enjoyed it. I still remember this from over a dozen years ago.
Get involved, most passengers are nice people in their world, but they are in your world, so take care of them...extra care if you have to.
;-)