First, we must realize that there are many people who believe that judges should be elected. Judges are elected in many states. People who believe that judges should be elected generally do so because they think that this is democratic. In this view, a democratic system ought to have judges that are directly selected by the people.
If we need to argue that judges should be appointed, we can base our argument on the idea that judges should be driven by their expert reading of the law. In this view, judges are not politicians who are in office to implement their own political agendas. Instead, they are experts who are in office in order to apply the law to given cases. If this is how we see judges, appointment is a superior form of judicial selection. It is more likely to get us judges who will base their rulings on the law, not judges who are going to make their rulings with one eye on getting reelected.
Thus, the main argument for appointing judges is that it will allow them to base their rulings on the law, not on what is likely to be popular.