My husband is a big believer in being prepared. At first I laughed it off, but over the years I have come to agree with him. I would rather be prepared and nothing go wrong, than unprepared and something serious happen.
I do not think that children should be scared or upset though, but I do think it makes sense to help prepare them. Since my son was very young we've practiced fire drills. We have always practiced different routes to get out of the house and to get to the meeting location outside. I think that preparing for any other disaster is the same as preparing for a fire. The kids don't have to be scared, just instructed on what to do for their safety.
And stocking up on a budget is really simple. Start small, adding one or two items at a time. When we started all we could do was $5 extra every grocery shopping trip. I bought a can or two of food each shopping trip and put it with our emergency supplies rather than with the regular groceries. Or I bought an extra flashlight, toilet paper, soap, or whatever was on the list. I just bought extra when they were on sale and used coupons. I bought blankets from the Goodwill, some clothing too, and a good sturdy pair of shoes for everyone. We also have "go bags" with a few essentials in them for each person.
We don't think it is a waste of money. On some show I watched a man said that when he was unemployed nearly a year all the things they had stockpiled were a life saver. They had intended the supplies to be for a natural disaster, but used them to survive unemployment instead. They used savings for bills like electric and mortgage instead of on food and such. They were able to survive almost a year of unemployment until he was able to find another job. What a blessing they had spent money in the past to prepare for the future!