Begonias do best with good drainage so the roots are never soaking in water, cut off from oxygen. There is more than one way to create a suitable growing medium in a terrarium, but generally speaking a loose, porous mix of equal parts peat moss (moisten before use) and perlite works well for begonias. Some mixes contain other components, such as sand or vermiculite, which is fine. It might take some experimentation to figure out what works best in your terrarium, but stay away from potting mixes that contain soil as they can hold too much water. Another option to experiment with is placing a layer of gravel or small pebbles under the growing medium to help with drainage. Plain green sphagnum moss is another common growing medium for terrarium begonias. Wet the moss with hot water and squeeze out the excess before placing it into the terrarium.
Begonias generally prefer shade, so keeping them out of direct sun is a cardinal rule of growing them in terrariums. Give them filtered light, such as from a north or east window in summer or a south or west window in winter. Fluorescent lights also work well. Leave them under lights for 12 to 14 hours per day. Experiment with artificial and natural light and location based on where you live and seasonal changes.
While begonias do not like their growing medium to be too wet and soggy, they do like high humidity. This is good news for the terrarium gardener, since a terrarium naturally keeps the moisture inside. Aim for a minimum of 50 percent humidity to keep your begonia alive, but research the specific variety of plant to find out the ideal humidity. Some types of begonia thrive between 60 and 80 percent humidity, while certain types do best with nearly 100 percent humidity. Mist the terrarium with water as frequently as needed to keep the humidity in the proper range. Aim the water spray at the bottom of the terrarium, rather than on the begonia foliage, as water on the leaves can encourage disease. Use a hygrometer to measure the humidity inside the terrarium.
Growing plants in a terrarium creates a unique fertilization situation. Because the growing medium will never be flushed out, the salts from fertilizer will stay in the growing medium and build up over time. For this reason fertilize very little, if at all. If you fertilize, use a light plant food diluted by half once a year in the spring or summer.