Do a web search for apple tree diseases, and you’ll likely come up with enough to scare you into never harvesting apples again! Rest assured, the vast majority of these are unlikely to ever occur, and just about anything that does occur can be solved with a few simple solutions.
Pruning is probably the number one preventative measure that can be done to maintain healthy apple trees. It promotes good airflow and circulation, which has a number of positive side effects. Many diseases, such as powdery mildew, are caused by humidity, which is increased by poor pruning. Increased circulation also ensures that if you use any sort of fungicide or pesticide, the spray will adequately coat the entire tree and fruits.
When fungus and diseases are found, it's important to prune away infected fruit and branches before it can spread, and to remove everything pruned from the area of the tree.
Fungicide coupled with pruning can go a long way to preventing any disease from infecting your apples or the tree itself. A decent “all purpose” spray is a sulphur spray. Used regularly starting when pink buds begin to appear, it will prevent many of the major diseases including powder mildew, apple scab and cedar-apple rust.
Between pruning and fungicides, your trees should remain healthy and bear plenty of fruit. If you're still concerned, consider looking into trees known to be disease resistant, such as Red Delicious Trees or Liberty Apple Trees.