EDITME
Babies
During a disaster or survival event, you will need to be resourceful and plan ahead as you will not want to pop down to the store for more diapers during the emergency.
Babies are not yet toilet trained, so they need either paper or cloth diapers to catch their waste. You can have a small supply of the correct size on hand, but when you run out, then what? Cloth diapers, if one can find them, can be washed and soaked by hand in bleach or ammonia and line dried. Yards of flannel can be cut and used for diapers.
Some cultures don't use diapers at all and teach the babies to cue when they are going to do the deed so they can hold them over hole or trench in the ground. (You know how slippery and wiggly babies can be so I am sure one must hold on tightly.)
Make sure you have adequate clothing to keep the growing baby cool and warm and protected from the sun, depending on season and temperature. That would include hats, very high-SFP value sunscreen, long sleeved shirts, jackets, socks and booties. The next size up in clothes is a good idea to have on hand.
Have enough formula, baby cereal and very clean, safe water set aside to mix with the baby's food, since that is the baby's nutrition source. Of course, if you are breastfeeding, the formula issue is solved and you only need to be sure you have enough water intake for yourself in order to make an adequate supply of milk.
Toddlers and Older Children
Make sure the children have several changes of clothes: underwear, t-shirts, pants or skirts, jackets, socks and shoes, hats, and sunscreen.
In an emergency situation, power may not be available and batteries will be used for more important things. Having low tech toys will provide a creative and energy outlet for the ones under your care.
Once the baby is walking and more adventurous, having toys that are safe for toddlers and the under-three set will keep them entertained and help you keep your sanity. Stacking rings, tupperware, blocks, Legos®, wooden spoons, and boxes are all great.
For older children, try children's clay, legos, Lincoln logs, coloring books, paper, crayons, construction paper, storybooks... any low tech thing that doesn't use batteries.
The older children will also be able to help out with minding and playing with the younger ones, and perhaps even some simple chores.
Safety
Watch your children at all times.
I repeat, watch your children at all times. That means actually seeing them. In an emergency situation, a moment's inattention can spell disaster. If you cannot watch them for any reason, ask someone else. Get a firm confirmation that they understand absolutely which child they are watching. Make sure you both know when you are expected to return, and that they know exactly where you are going. Tell them what to do if for some reason you are not back at the agreed upon time.
If you must take your eyes off of your children for a short period of time, be sure they are in a well-ventilated, not too hot or cold, contained space so they cannot wander off. If there is no such place available, consider if it is safe to use something to anchor them to the spot, like tying a rope to their belt. BE OVERLY CAUTIOUS IF YOU DO THIS. Make sure there is no way your good intentions can in any way harm them, like getting entangled and choking. It is FAR BETTER to watch them, or have someone else watch them.
Plan ahead, and make sure your older children understand where they are to meet you if you get separated. If your older children are watching your younger children, make sure they understand that they cannot do anything else until you return.
Follow all the rules of safety you normally do, but with a heightened sense of awareness. They are counting on you to protect them.