IS CRIME a serious problem in the District? Of course. From 1978 through 1981, the violent-crime rate increased 62 percent, and the property crime rate 38 percent -- growth rates more than twice the average for large cities and substantially above those in neighboring counties. Figures for recent months suggest that the explosive growth has stopped, but it's too early to be sure. And even if the growth stops, the absolute levels are far too high. What can a mayor really do about it? And how should the candidates' own positions be judged?

The great forces in crime statistics are economics and changes in the makeup of the population. So one ought to be very careful about blaming a mayor when crime increases, and just as careful about giving credit if crime slackens: sometimes it is only the result of a lucky demographic break. The valid test is whether a mayor makes the kind of effort called for by the magnitude of the problem, in light of the competition for money and attention. And the efforts ought to demonstrate creativity and competence in three areas: prevention, convictions and rehabilitation.

Mr. Barry's police budgets have barely kept up with inflation. He is increasing the size of the police force very reluctantly, and only at the shrill insistence of the overseers on Capitol Hill. But these are simplistic measures of commitment, since the most effective approach may be better use of what you've got rather than more money. It's hard to prove that a few more cops or cars would really make much of a difference, and none of the candidates seems to be calling for a massive infusion of money or personnel. The mayor has, on the other hand, presided over a major improvement in the relationship between the police department and the community -- a critical accomplishment.

On the prevention front, Mr. Barry points to his general support for summer jobs, housing and social programs (records his opponents also criticize, of course), on the theory that economic woes feed crime. But these are not specifically directed at crime prevention. He has pursued a number of modest but important initiatives, such as more street lighting in high-crime areas and assistance to private groups trying to improve security in housing developments for senior citizens. The most promising measure is the Neighborhood Watch Program of local volunteers -- a good idea, but one which the District lags behind several other jurisdictions in pushing.

On the enforcement front, the mayor points to his recent efforts to crack down on drug dealers, but more than 5,000 arrests have apparently produced fewer than 60 felony convictions. The low figure could be due to poor police work or to failings of courts and prosecutors outside the mayor's control, or both. Moreover, candidate John Ray fairly points out that Mr. Barry came belatedly to focus on the drug problem as a source of crime. The mayor cannot be said to have thrown himself into this particular battle during the past three years. The item Mr. Barry places first on his list of accomplishments in this area is his election-year appointment of a blue-ribbon committee of 100 to study the crime problem.

Mr. Ray has made crime the centerpiece of his campaign. He has set forth a five-point program, the key elements of which are combating drugs; improved job programs for inmates in the hope that they will make them less likely to commit crimes when they get out; community work programs and other alternative sentences for nonviolent first offenders, and mandatory minimum sentences for drug dealers and those who commit violent crimes with firearms. Much of this is thoughtful, although mandatory sentencing has serious pitfalls.

Mrs. Harris, by contrast, emphasizes the lost opportunities -- things Mr. Barry could and should have done to improve management and coordinate the several agencies with a potential role in addressing the crime problem. For example, she proposes a victim- witness assistance program to provide supportive services (health, counseling, transportation, household security, and so forth) and thereby not only be compassionate but increase the chances of getting the testimony necessary for convictions. She agrees with Mr. Ray on improved rehabilitation efforts at Lorton. Mrs. Harris also urges increased attention to technical assistance to help citizen crime watch programs, and a more aggressive effort to increase foot patrols. Actually, a number of Mrs. Harris' notions are already in place -- she generally urges "more" or "faster," rather than very different policies. Some of her other proposals are a bit implausible. Do we really need a Community Anti-Crime Assistance Corporation to help community groups combat crime? Can the mayor really be expected to provide leadership for the Superior Court and the U.S. attorney's office, both independent institutions?

Mrs. Jarvis has not made crime a major area of attention for her campaign, preferring to concentrate her limited campaign resources on economic development and housing, her areas of greatest involvement through her role on that council committee. She does, however, share with the other challengers a judgment that the mayor has squandered some opportunities at a time of crisis.

If there is to be any progress in the next administration, then, the next mayor -- even if it should be the present one -- must make a heavier investment of personal leadership to combat the rise in crime.