My wife comes from a large family, and as such, we have 12 nephews and nieces to take pictures of every time we visit Mexico.
I find that the DoF is a factor of both the aperture, but also the focus length. What I found works best is my 50mm f/1.8 lens, opened up at around f/2.8. With 50mm, I find that the DoF isn't too restrictive. You can have a subject move a few feet forward or behind, and the image still comes out nicely. I don't own any fancy USM lenses, so I mostly use manual focus, and that doesn't cause me any troubles. But that said, I usually try to "trap" the kids so they stay within the DoF. Good examples are to move the table near a wall when taking pictures of birthday cakes, that way you can keep their movement in the direction that doesn't affect your DoF. Or position yourself aiming somewhere you see them pass often, just waiting for them to pass in your DoF.
Last week-end I had some fun at a butterfly exposition where you can walk in a large room full of butterflies. That reminded me taking photos of the little ones. They move fast, and they move erratically. In sports photography, you have a general idea where the action is going to be next. In wildlife photography, depending on the subject, it can be much harder to predict their patterns. There are no rules, there are no nets, no ball/puck/whatever, just an infinite 3-D universe where things can happen.
But also, just like wildlife photography, you have to set your expectations correctly. If the kids are sitting still, you can get a good success rate on keeper photos, but if they're playing around like crazy, then don't set your expectations too high, maybe 1/10, or even 1/30 photos will be keepers.
And alternatively, you can take videos in 1080p, freeze whichever frame you like best, and print it out to a 4x6.
Last subject I want to touch is lighting. In professional movies, there are very strict rules about how long a baby can be exposed to intense lights. Babies and kids are much more sensitive than we are, so try to use special lighting as little as possible. If you can make do with ambient light, even the better. The flash can really hurt their eyes, and if you're badlucky, can even cause permanent damage. The younger they are, the more careful you have to be with light.
Cameras: Canon EOS 6D, Canon EOS Rebel T3i, Canon EOS Rebel T2i, Canon S90
Lenses: Tamron: SP 24-70mm F/2.8 Di VC USD, SP 70-300mm F/4-5.6 Di VC USD, Rokinon: 8mm Fisheye cine, Canon: EF-S 18-55mm f/3.5-5.6 IS Lens, 75-300mm f/4.0-5.6 III, and EF 50mm f/1.8 II
Retired camera: Fujifilm Finepix s700