To make the drain pipe on the sink bigger is difficult, as this stuff is usually certain standard size and hard to modify.
What you could try, however, is to limit the water flow. Under the sink you should have valves to shut off the water (cold/hot?) going to the faucet, in case it leaks and has to be replaced. Use the valves to reduce the water flow to a point, that the drain can handle it if its running fully open unobserved.
And brief the children to use faucets with care - not leave it running if nobody is in the room ect.
A additional precaution could be a leak alarm. A little device that you place on the floor in a place prone to flooding. If it gets wet, it makes a hell of a noise, like a smoke alarm. That way its only a little bit of water that gets spilled. They dont cost much, may be 10 bucks or so.