Corn snakes should have constant access to clean drinking water. While the size of the water dish is not necessarily crucial, some snakes enjoy soaking their entire bodies to soften their skin prior to and during shedding, although this is not essential to their captive success or health.
Larger water receptacles offer temporary safe refuge in the event of cage overheating.
The water should be routinely changed at least twice a week, and immediately if soiled or cloudy-looking. City tap water often contains chlorine and fluoride that could be in relatively harmful proportions for small snakes (my unscientific opinion). Filtered drinking water (affordably available at all grocery stores) is recommended, but do not use distilled water as it lacks valuable minerals essential for proper nutrition in your pet.
Home water softeners add minerals and chemicals such as sodium (or potassium), and are therefore not recommended. Never add water to existing water levels. Dump out the old water, clean or replace the water receptacle, and replenish with fresh water.
I have not attributed city tap water in the cities where I have lived to any medical conditions of my adult corns and have never actually attributed any neonatal snake death to the chemical contents of city tap water.