The design of any concrete structure must consider loadings and environmental conditions. This includes drying shrinkage of the concrete, and daily and seasonal thermal changes that introduce stresses within the concrete sections. Domes are thin-shell concrete structures and stress concentrations are expected around any openings through the shell. Typically, the design engineer will provide additional reinforcing bars around openings and especially at corners of openings to accommodate the buildup of stresses in these locations. Also, chain-link fence is not considered as acceptable concrete reinforcement, as it cannot accept tension in a straight orientation within the concrete. Proper concrete reinforcement is either deformed reinforcing bars or steel mesh with smooth wires laid out in an orthogonal pattern.
You should have a professional engineer with experience in concrete shell design and construction evaluate your dome home for structural integrity. If the cracking is determined to not affect the structural integrity, epoxy or polyurethane grout injection is routinely used to seal cracks in concrete.