It is some kind of damping-off fungus that probably came from your basement. These funguses will usually die off in the heat and sunlight. In the meantime, trim off the affected leaves. I will usually not trim leaves that are just beginning to get infected if I think I might be pruning back too much -- the plant needs some leaves to photosynthesize. The new growth usually doesn't get infected too much. This is not powdery mildew, which kills cucumbers within a week or so of being visible. Most of your plants should survive without much of a setback. Cucumbers are really finicky though, especially about being cold at all.

Container growing with bagged soils almost always creates a micronutrient deficiency of some sort. Be sure to include a high calcium amendment in the soil mix and a rock dust product such as Azomite to provide those nutrients. Sometimes it's an issue with uptaking those nutrients and epsom salt can sometimes help with this as well as regular watering, don't let containers dry out or it's over....

Since the Op mentioned that they "got this cucumber plant recently", it appears the plants may not have been hardened off sufficiently prior to transplant. I would expect to see far more damage had they been exposed to frost, especially for young seedlings; the newest leaves appear to be healthy. The only cure necessary may be time... that, and perhaps thinning the clump to reduce crowding. And just as an aside, I concur with the post just above. Anyone who has gardened for any length of time knows what the USDA hardiness zones mean, and what they don't mean... so give it a rest, and let's stick to answering the OP's. I wish the USDA would switch to a system that is more indicative of overall climate (such as the Sunset zones) and put an end to the arguments.

You never want to let cukes dry out between waterings. At best you'll end up with very bitter cucumbers. Cukes require consistently moist, not wet but moist, soil. Did you harden them off well before moving them out to the garden? If not that would explain some of the damage as nothing can go straight from inside out to the garden without a gradual period of adjustment to full sun and wind first. So are these symptoms on all the leaves, even the new ones, or just the older leaves? Does it appear to be spreading up the plant? Does it rub off with your fingers? Is it more yellow than white or brown? If so then it is Downey Mildew as opposed to Powdery Mildew.