Washington's $1 billion tourism industry is starting the summer with a slump as family vacationers stay away in droves because of tight gasoline suppies here.

Not all segments of the tourist economy feeling the pinch. Hotel occupancy rates - buttressed by heavy air and rail travel - remain strong.

But there are indications that people who drive to Washington to see the sights - and who are the mainstay of the summer tourist trade - are balking as the word spreads about the city's misery gasoline pumps.

"Obviously we are concerned, because the family vacation time is rapidly approaching, and that's where we anticipate to be hurt the most," said Austin Kenny, director of the convention and visitors bureau for the metropolitan area.

Some examples:

In the last two weeks, Tourmobile, the bus company that serves National Park Service attractions, has recorded a deline in ridership of nearly 20,000 persons compared with the same two weeks last year.

Park Service officials said 25,000 fewer persons ascended the washington monument last month than during May 1978.

The operators of George Washington's family home at Mount Vernon say their attendance figures have dropped by 12 percent Representing 6,000 persons - duringthe first two weeks in June.

Even the White House tour, which has perphaps the strongest drawing power of all Washington sites, is barely keeping pace with last year's figures.

For the period June 8-14, the White House reported 28,730 visitors compared with 28,100 last year. However, that figure may represent an actual loss in attendance since the White House was open one extra day this year during the period counted, a spokesman said.

But while some tourist businesses and attractions suffer, others prosper as usual. The best example is hotel occupancy rates, which are increasing in most cases. The reason, officials say, is that most visitors to washington - 75 to 80 percent - travel here by plane or train. And their numbers are increasing.

However, June marks the hotel summer season. School is out, convention business declines and car-driving family tourists become a mucg greater economic factor, officials say.

Indeed, hotels and motels already are reporting noticeable declines in weekend bookings, traditionally dependent on family vacationers.

"Obviously if this shortage continues, the hotel industry is likely to suffer said Len Hickman, director of the Washington Hotel Association, which represents 43 big hotels in the District with $200 million in annual sales.

"Generally, they [hotel managers] are telling me they are getting more 'no shows' on the weekend. Some members are telling me their July and August does not look good for bookings," Hickman said. "We are not alarmed, but we are very concerned."

In general, a survey of Washington area tourist business showed that those that depend most on car-driving visitors are experiencing stagnation or decline and those that depend on visitors who come by air or rail service are holding steady or increasing.

"They're [tourist] staying away," said Tom Mack, Tourmobile's general manager, who added that the "larger percentage" of his customers come to Washington by car.

The slump apparently extends beyond tourist sensitive areas. A popular barbecue restaurant in Rockville reported Saturday that its business had dropped off by more than half.

At the same time, ridership on Washingtonhs major charter bus service, Gold Line, which transports 5,000 to 6,000 tourists a day, has increased 10 percent over last year.

The reason, according to Gold Line superintendent Al Chaulker, is that the bus charter service is more dependent on organized travel agency tours that come to town via plane or train. And, Chaulker said, Gold Line solicits its business in hotel lobbies where occupancies are high and demand is growing for organized tours.

"We're so darned busy we have to turn some work away," Chaulker said.

Some campgrounds also are experiencing a decline in business. Norman Gurevich, who operates Cherry Hill Campcity in College Park, said the normal occupancy for his 240-space campground is 90 to 95 percent for this time of year. "Now we're running about 85 percent," he said. He also said there is a definite decrease in the number of large campers on the road.

Moreover, Gurevich said that more campers are carrying extra fuel tanks and 'more people are traveling with tents."

Gurevich said a major factor in the decline of his business this year "is that we can't tell our customers when these [gas] stations are going to be open."

To help head off any anticipated decline in hotel bookings, Hickman said the association is launching an advertising campaign this week to tell potential tourists in other cities that "Washington has solved the energy crisis."

"We're telling people who read the papers that once you get here . . . you don't need an automobile to see the sights, which basically are in a 20-block area," Hickman said.

Outlying hotels and motels may feel the effects of the gasoline shortage much quicker, officials say.

"We're starting to see a tapering off of weekend business, but nothing substantial at this point," said Jay Mahan, the general manager of the Sheraton Hotel near the Pentagon.

It was the same at the Twin Bridges Marriott near National Airport. "We've had a few cancellations, 1 or 2 percent on the our vans back and forth to the airport."

Another suburban motel cut its airport shuttle service in half because of the difficulty in finding pasoline for its three vans.

Robert McGrail, an executive with Quality Inns International, which operates 12 motels in the metropolitan area , said that his chain's occupancy 12 motels in the metropolitan area, said that his chain's occupancy figures "are generally ahead of last year."

"The thing that hurts more than anything else is the fear . . . We've received cancellations from people afraid they would not be able to get gas," McGrail said.

Hickman's member hotels in the District reported 85.4 percent occupancy during May, a 1.3 percent increase over May of 1978.

One way that Sheraton's Mahan is monitoring the effect of the sortage on his tourist business is by keeping an eye on his parking garage. Every morning at 5:30 a.m. Mahan sends one of his employes into the garage to count cars. What he has found is that the number of cars there has dropped as much as 15 percent from last year. But the hotel's occupancy "is beating last year by about 4.2 percent," Mahan said.

That tells him, he said, that "people are still coming in. Just not as many of them are driving."