The U.S. Marshals Service, an agency whose responsibilities range from rounding up fugitive criminals to providing protection in federal courtrooms, is running out of money.

Director Stanley E. Morris said the agency has moved into low gear since last March when, like every other agency in Washington, it was hit by the Gramm-Rudman-Hollings budget-balancing law, and absorbed a cut of 4.3 percent.

Morris had expected things to return to normal after President Reagan proposed a $12 million increase for fiscal 1987 over the Marshals' pre-Gramm-Rudman-Hollings 1986 appropriation of about $150 million. But at a recent hearing, the House Judiciary Committee rejected the proposal without discussion.

Staff sources said panel members believed the Marshals Service should not get a budget increase when so many other agencies face cuts. Morris is hardly the only agency head in Washington crying poor as legislators get down to business on the 1987 budget.

But Morris insists that his agency will not be able to adequately fulfill its duties without a funding increase.

One problem, Morris said, is the increasing number of dangerous trials that require extraordinary security -- particularly those of Colombian drug dealers being extradited to the United States.

"It's an increasingly difficult job to dispense justice when the defendants are prepared to kill judges and anyone else who gets in their way," he said.

He pointed to the case of Barry Seale, a former drug trafficker turned government informant who was killed by a Colombian hit squad last February as he was reporting to a Baton Rouge, La., halfway house where he had been sentenced to spend nights for six months.

Morris said the alleged hit men, who were captured, have large amounts of cash at their disposal and have twice tried to buy their way out of jail.

Another Marshals Service nightmare is the possible trial of Colombian drug kingpin Jorge Ochoa. Ochoa is in prison in Spain, and the United States is seeking his extradition for trial in Miami.

Referring to Colombian drug traffickers, Morris said, "They just don't have the same ingrained respect for the judicial process that we may expect in this country."

A Marshals Service study in fiscal 1985 showed that in 10 of the nation's 94 judicial districts, Marshals using metal detectors found more than 20,000 weapons, including 3,000 firearms and 10,000 knives, at courthouse entrances. Although only a small percentage of those weapon discoveries resulted in arrests, Morris said he finds the statistics disturbing.

During that same period, a nationalist group launched a rocket attack on the Marshals' office at the federal courthouse in San Juan, Puerto Rico.

Since the Gramm-Rudman-Hollings cut went into effect last March, Morris said, the Marshals Service has started to cut back in areas not affecting security.

For example, fewer people are assigned to searches for federal fugitives. "That's one thing we can control," he said, adding that unless the president's budget is restored, the projected number of fugitives caught by marshals next year will drop from 6,000 to 3,000.

Morris said he also is asking judges to postpone trials and to start them later in the day so the agency can avoid paying overtime. Although that saves money for the Marshals Service, Morris acknowledged that it shifts costs to the judicial system.

The extra $12 million the administration had proposed included $6 million for security in high-risk trials.

Morris said he doesn't know what he'll do if called upon to provide security for a trial involving a highly visible, dangerous criminal.

"The problem is, we can't plan for these things. We don't control our work. We're basically a service organization that serves the needs of judges, the needs of investigators," Morris said. "When Rudy Rudolph W. Giuliani, U.S. Attorney in New York wants to try the five organized crime families, we've got to provide security whether we want to or not."