Please tell me what is the safest atrial septal defect closure method?

It depends? An ASD is simply a hole in the wall between the right and left atria (upper chambers of the heart) many, most can be closed in the cath lab by a pediatric interventional cardiologist or an adult cardiologist trained in structural heart disease. There are two fda approved devices (amplatzer septal occluded and the gore helex septal occluded). Some holes need to be closed by an open heart surgery.
Surgery. While there are multiple methods available, the best long and short term results are with surgery. Direct or patch closure of the defect is safe, reproducible, and durable.