Is it safe to reuse steamer water? My youngest is a fussy eater, and we have problems getting him to eat vegetables.  Recently we've tried using a wide steamer and he likes it, because its wide enough to keep all the carrots from touching while cooking (yes - that's his reasoning)

However he also only eats carrots and broccoli when there's a trace of sugar in the steaming water - a teaspoon or so.

My wife is happy to leave the water in the bottom pot and let it cool, then reheat it the next evening.  I feel this is a bad idea but can't put my finger on why....  The water boils so it reaches temperatures that should kill anything bad.

What have we missed ?
Your wife probably thinks "It's just water and thoroughly boiled, so all is well.". Unfortunately, it is not.

If you look into your pot after the first use, you'll notice that the water contains traces of the vegetable, meaning it technically falls under the "food that becomes unsafe after 2/4 hours at 40-140 °F (4-60 °C)" category and should be refrigerated quickly or discarded. More on this on our canonical post on the topic. 

Perhaps you argue that cooking will kill any "nasties" that might have grown since the day before, but while this may be true for most bacteria and fungi, it does not work for some of the toxins they produce. Boiling does not make unsafe food safe. 

Weighing the cost of a bit of water and sugar agains the the risk of a child's health, I'd dump it and clean the pot every time.
You could put the pot in the refrigerator, but must heat and cool it as quickly as possible (keep in mind that the time is cumulative) - which sounds to me like a lot of excess effort and takes up a lot of refrigerator space.