The two sides meet on Sunday in the first match at Anfield since the Hillsborough Independent Panel's report was published last week.
There is a history of fierce rivalry between the clubs, but the Red Devils skipper will join Liverpool captain Steven Gerrard in releasing 96 red balloons before kick-off in a tribute to the victims, and Vidic hopes the supporters will follow suit in paying their respects.
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"I am aware of the sensitivity of the day," he told reporters. "Everybody is. There is a lot of talk in the papers and the media. I think our fans will respond well.
"The whole country is on test in this one. All over the world they will be watching this game. It is probably the biggest derby in the world and we have to show we are capable of keeping a good atmosphere and being a good example to the kids.
"There is a lot of history with these two clubs. I think we should respect each other because we are big clubs and clubs that are respected in the world. We have to be on top of the bad situation and behave well and, I would say, be an example."
A minority of United fans were heard making anti-Liverpool chants during last Saturday’s win over Wigan, sparking fears over their behaviour on Sunday, but Serbia international Vidic is hopeful that the occasion will be remembered for positive reasons.
"Football is important but it is never more important than life, any life," he added. "It doesn't matter if it's in England or Serbia or any part of the world.
"I think most important is that we have to respect any life and I think that we will do it on that day."