It sounds like your potting mix is too wet/saturated, which is the real problem (not the fungus, which is merely taking advantage of favorable conditions). Too much water and not enough air in the root zone equals unhappy plants. My guess is that you're using a commercial bagged potting mix of some kind? There's not much you can do to enhance drainage at this point, so, should the rainy weather continue, you might want to try to cover them if it's feasible.

Thanks so much for your help! And yup, I used Miracle Gro potting mix. You clarified the problem for me, good to know that it's not the mold or fungus. I am putting trash bags over them as soon as I type this, it's raining yet again. Much appreciated!

What sort of potting mix are you using, how much sun are you getting and is the water from the wall/floors/balconies above you falling into your pots? There is a big difference between rain and a constant stream of water cascading down the side of a building into your pots, I had to move some of mine for that very reason. I also grow on a balcony in NYC and I havenâÂÂt really had a problem with the water this year but that could be because I get a good 7 hours of direct sunlight to help dry them out.