CARING FOR indigent patients is going to cost Washington's hospitals more than $200 million this year, and those costs are rising fast. The hospitals are unable to carry the burden, and it's the city's responsibility to work toward a solution. Sharon Pratt Dixon, the mayor-elect, is already being drawn into the issue.

By the last precise count in 1987, 114,000 people were living in the District of Columbia with no health insurance -- just more than one-fifth of the population. Most of them belonged to families in which at least one person had a job, but a lot of the jobs in this city -- one out of six -- carry no health benefits. One-fourth of this uninsured population was children.

Here in this city, as in the nation, the cost of universal health insurance places it out of reach for the present. But some incremental progress toward that ideal is both possible and necessary. The city's Indigent Health Care Public Policy Action Forum is working on cost estimates for various possible definitions of basic health coverage. One approach is to begin with a carefully defined part of the population -- for example, children. That route is particularly appealing because federal law is now expanding Medicaid coverage of children.

Whatever is done will be expensive, and the hard part of the exercise is finding ways to finance it. Virginia is going to try to do it with a 2 percent tax on hospital revenues -- in effect, re-establishing by law the cushion that hospitals used to put into all their bills to cover charity cases before the insurers' cost controls squeezed it out. Unfortunately, as the Rivlin Commission has pointed out, 2 percent won't suffice here in the District because of the much higher concentrations of indigence. A higher tax in the city would make the city hospitals unacceptably expensive in comparison with the suburban hospitals with which they increasingly compete for patients.

Financing will probably come from a package of small and tentative sources. One possibility is a tax on health insurance. Another is a requirement that employers provide health insurance at least to all full-time employees.

The progress that any local or state government can make in the absence of national health insurance is limited. But Mayor-elect Dixon and the city are going to have to press those limits. Otherwise the costs of caring for the uninsured poor will undermine the quality of all the city's hospitals for all the city's people, indigent and affluent alike.