You need to know about damping off - it's a fungus that hits at the soil line. The first signs of it may be difficult to discern, as it is a brown growth on the surface of the soil - brown growth + brown soil = hard to see. But, if you carefully watch for it, you'll notice a difference in the soil surface texture. The next sign is a softness of the stem of the plant at the soil line. Eventually, the stem becomes so weak, the plant falls over. Usually this results in the death of the plant. Bottom-watering is one way to prevent it - making sure the water does not reach to the top of the soil. Running a fan on the plants both helps to simulate wind, strengthening the stems of the plants, and also to dry out the surface, to help prevent the fungus from growing. But yes, it can still happen. You can scrape the fungus off, then spray the soil surface with a mix of hydrogen peroxide and water to help kill any remaining fungus. Ideally, lightly spray the soil surface from time to time to prevent it in the first place, too.
Damping off most often occurs with top-watering, and younger plants are more susceptible to it than older plants, though older plants can be killed by it, too. As noted by others, regardless of your watering method, make sure your containers have plenty of drainage. I use Solo cups and cut fairly big holes in them, so they're fine to use. I fill a pot or the sink with an appropriate amount of water, set the cup with plant in there about a minute, then pull the cup out, letting it drain until it no longer drips. Keep in mind that pepper plants tend to like moist but not wet conditions - it is best to let the leaves start to droop just a tad before watering again.