I bottom water. Originally I'd let the trays sit in water until they had their fill, but this would leave the soil very wet, even up top, and as a result I had algae growth.
They need a wet environment to grow as well as light. If you're bottom watering keep an eye on the cells and stop when you see water nearing the top so it stay wet for days. You could put a layer of something else at the top that doesn't hold water as well, maybe stone or sand.
The humidity dome is definitely going to help the algae, but if it's necessary for your peppers it's going to have to remain a double edged sword. I'd recommend a fan later on to keep the air moving.
I've had some success mixing baking soda in water and spraying the algae. It's supposed to be to basic for it to grow. Try not to get it on the plants (though it won't hurt them much). If you have enough room just sprinkle a tiny bit on there or make a slurry with more baking soda, less water and apply it carefully with a q tip