Is it safe to reuse steamer water?
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.