MAYOR Sharon Pratt Dixon, who is working to free this city from a potentially devastating financial burden, is already hearing from every corner of the city about what she proposes to cut from the budgets. Some of the most intense pain is being felt by groups that have enjoyed not only great public favor but also fairly good resources over the years. A few nights ago, it was the local theater and arts organizations that rallied for the mayor's financial attention at a kind of town meeting. The subject: money for the D.C. Commission for the Arts and Humanities. The object: keeping any budget cuts to a minimum.

The numbers got confusing, and so did the official responses. To some, it seemed as if Mayor Dixon was backing off an earlier budget cut proposed for the commission -- and she was. To others, it seemed she was making a cut even larger than the sizable one proposed by Mayor Barry beforehand -- and she was. By the end hardly anyone was sure whether the mayor had modified her position in accord with the group's wishes or not.

If you start with the first agency-by-agency cuts she proposed after a day in office, the mayor has indeed left a little more than that in the commission's current budget. But she has also proposed cutting a little more -- about 24 percent -- than the cuts Mr. Barry had proposed -- about 23 percent -- but not made. So the cut being made is large, and the mayor is holding firm now. We think this is an unfortunate necessity, in fact the right thing to do. Every constituency, some with acute social and economic problems and needs, is going to be affected. In these circumstances, you cannot justify making a large exception for arts funding. These programs do become more deserving of money from private sources as a result. Businesses and foundations, of course, aren't doing so well themselves, but there are some that might well take a second look at their commitment to the arts here.

Mayor Dixon cannot afford to buckle every time pressure is put on her to undo her plans. The money just isn't there -- and as far as anyone can see, neither is any willingness on the part of taxpayers to pay more. The federal government does owe more than it has been paying, to be sure; but extracting any part of this is extremely difficult until the city can demonstrate an ability to make its own painful cuts. These economic conditions happen to have fallen hard on Mrs. Dixon's watch, and she needs to be encouraged and supported in dealing with them.