hey, charlie! :--)
this is not only a training problem (meaning the dog is being bad),
it is a management problem - meaning humans provide opportunities for the dog to do the un-wanted behavior.
here are my suggs to eliminate opps:
1) baby-gate the dog away from the food area,
2) do NOT allow kids to walk around the house with food in hands -
they must sit in the kitchen or dining area, on a high-chair or chair,
and the dog should *not* be in the room, so that when (not if!)
the child throws or drops food, the dog cannot possibly get to it.
3) set the dog up with tether stations around the house,
by either screwing an eye-bolt into the baseboard + clipping a double-ended bike cable to the eye at one end,
+ dogs collar at the other...
OR make a portable tether by taking a 2-ft length of 2 x 4, installing the eye-bolt in the middle of a wide-side,
and clip the bike-cable to the eye-bolt. THEN put the 2 x 4 section behind a door on the floor, slide the CABLE
under the door, and Close the door! Ta-da, U have a tether-station, and a portable tether...
Any door in the house can now become a station for the dog, where the dog is under human supervision,
not alone, but cannot get to the fallen food, toy in Babys hand, etc.
the cable-length in these restraints should be 18 to 24-inches, Max.
clip the free end of the cable to the dogs collar, and U are in business~!
U can lay down a mat at the tether-station, the dog can have chew-toys, a stuffed Kong with part of their meal in it, either damp/frozen or mixed with low-fat grated-Mozzarella, and melted into a single hunk in the microwave for 20-seconds...
either freezing or welding together makes the food in the Kong harder to extract, and it lasts longer.
4) if U do not like the idea of the tether-stations...
try an umbilical cord!
take a 6-ft long leash, and either run it around Ur waist, then drop the clip thru the wrist-loop and
lip the free-end to the dogs collar...
OR -- take the wrist-loop, run a sturdy BELT thru the loop and then buckle the belt.
slide the wrist-loop to whichever side U want the dog on, clip the free-end to the dogs collar.
Now wherever U go, the dog goes... and the dog cannot possibly snatch anything + run off with it,
they are right under Ur ALL-SEEING Eye,
one or the other of these, or a combination, should cover the period until the children are old-enuf
to be TALL enuf that the dog cannot snatch items from infant hands...
and eventually, the kids ** will ** stop dribbling food, everywhere they go, LOL,
and it will stay in the bowl, on the plate, etc.
Dogs are opportunists... The Dog Rule is that if No One is in actual POSSESSION
of food, or a bone, toy, etc, then the item is Up For Grabs. by being careful
NOT to leave items lying-about where they can be snatched, Ex - on the counter,
on the table with no on in the room, etc --
we can avoid presenting opps for the dog to profit by our poor management.
the more-often the dog finds yummy bits lying-around, the more times We Are Rewarding
The Very Behavior That We Claim To Dislike, And Are Maintaining That Un-Wanted Behavior.
sad, but true...
be the change U want to see in the world, LOL...
put Ur stuff away! 
wouldn;t Ur mother love to hear that, LOL?
manage first... Then train. cheers,
--- terry