When I read this question, I thought not of women and minorities but of the characters that are on TV. For instance, the Judge Judy types or the wild caricatures on TV legal comedy-dramas. I suppose if people considered judges to be more interesting, they might consider courts to be less stuffy. A lack of stuffiness does not necessarily equal fairness though. A judge's job is to keep things legally fair, which is not necessarily the common public's perception of fair.
To go along with krishna's post, as these judges views and beliefs change with their differing backgrounds and the period of time they are in, would their opinions not more closely reflect the views and beliefs of today's society in general.
I'll simply make an assumption of what you mean by nontraditional, as in those judges who can make rulings individualized to the case presented, as opposed to operating under mandatory guidelines, or those judges who sentence people outside of the normal means of punishment (jail, probation, etc.). Tribal judges, for example, have sometimes banished individuals from their tribe, or in one case I know of, even banished two youthful offenders to an island for years. Certainly non-traditional (or maybe, in that case extremely traditional!).
But in terms of the view of the courts, I think it could aid in the perception of fairness the public holds of the institution, but at the same time, it might not, because of inconsistency between judges and rulings.
What do you mean by "nontraditional?" Do you mean women and minorities? Do you mean judges who weren't lawyers? Do you mean judges who rule based on their own ideas rather than the law? The answer really depends on how you are defining nontraditional.
If you mean women or minorities or non-lawyers, I think that these would reduce the perception of fairness. If people felt like judges were being picked on some basis other than merit, they would have less confidence in the courts' fairness and competence. Some people, however, would probably feel the courts were more fair because they would no longer feel like the courts were slanted in favor of rich, well-educated white men.
I will answer this question in a general about the influence of the beliefs and values of the judges on the judgements passed by them, without grouping judges in any classes such as traditional and nontraditional. This way, I will be able to answer the question satisfactorily without getting into the controversy of meaning of "nontraditional".
People who frame the laws try to make them as clear and explicit as possible. However, trying to make laws too detailed and too explicit can make them very cumbersome and rigid. Therefore all good law have some scope for different interpretation by people who implement these laws. For example, a law may prescribe that the guilt of a person must be proved in a court of law beyond all reasonable doubts. However it is not possible to describe in detail all the conditions of being reasonable that will be applicable in all the cases that are likely to tried in the courts. Thus the judges do have to often use some discretion in giving judgements.
The nature of discretion of any judge is very much influenced by the beliefs and value system of the judge. These beliefs and value systems of judges are in turn very much influenced by their background and past experience. For example, a judge who has been brought up in a culture where personal freedom is highly valued is likely to rule more in favour of personal freedom. Similarly, a judge who has seen the suffering of a cancer patient within his or her family is likely to be more sympathetic to cancer patients. In this way the view of fairness of courts is very much influenced by the beliefs and values of the judges.
Further, there are some general trend of changes in beliefs and values of the people in a given society. This applies equally to the judges in the society. Therefore over a period as older judges retire and new judges from the younger generation are appointed there is usually a gradual change in the view of fairness of the court. We can attribute such change to the change in general characteristics of the judges.