I primarily use glass containers. I figure glass is less likely to leach anything into the water & reduce the effectiveness of the distilled water as a solvent. I also have an affinity for glass containers and find them satisfying to work with. I use containers I purchased from a kitchen store that I believe are intended to be used to store spices. They look like this:
They're water tight and the rubber seals come in different colors to help differentiate one from the other. I put a paper label on the top where I write the type of coins, the count, and the batch number (row number in the spreadsheet I use to track cleanings, soak time, and last water change date).
What type of containers do you use?