First Baptist Church of Washington, a stately neo-Gothic structure at 16th and O Streets NW with a congregation rich in Southern manners, has changed noticeably since Inauguration Day five months ago.

It is there on Sundays that a presidential motorcade arrives to deposit President and Mrs. Carter, Amy and Chip and Caron Carter for Sunday school and worship service.

They try to blend unnoticed into First Baptist, whose familiar folkways offer them a feeling of home. But the accompanying press pool, Secret Service agents and gawking tourists - some dressed in dungarees and T-shirts - have made normal worship almost impossible for the Carters.

Half-empty before, the church is now packed every week that the chief executive attends, and another 200 spectators wait outside to wave at members of the First Family as they come and go.

The prayers always mention the President and Mrs. Carter and the bulletin reminds visitors to stay in their seats while they leave by the side door.

As Carter walked in late to his Sunday school class recently, all eyes darted to the President, who said pleasantly, "Good morning." When the first question needed to be answered, teacher Fred Gregg, a Tennesseean, glanced automatically to Carter, and the chief executive promptly responded.

"He knows he can't be anonymous but he insists on no fuss and feathers for him," said associate pastor Dr. Charles Sanks.

The First Baptist congregation, which lobbied hard for Carter's membership, has guarded its privilege zealously. Absolutely no staff member will reveal which Sundays the First Family is expected or when the President will teach Sunday school, which he has been doing monthly. "The White House knows they can trust us," said Dr. Sanks.

Somehow, Carter watchers from Washignton to the Far West find out the President's schedule anyway, and on the days he is expected at church, spectators swarm into First Baptist in such throngs that members now have to arrive early a half-hour or more to find a seat.

Just seating the congregation so as not to interrupt the decorous and staid worship service can be "murder," sighed one usher. "We hope we can keep it in stride."

At the last minute, busloads of 30 or 40 visitors often ple n, and the ushers have been instructed to seat them downstairs in a special room rather than setting them free in the packed main sanctuary. The word is out that bus groups receive this treatment, and consequently, last-minute arrivals have dwindled.

Also, there is the problem of cameras, which signs on the door forbid. Not everyone notices, however, and the ushers have firmly turned visitors away when they refused to relinquish their camera equipment temporarily.

Even more irritating to First Baptist members, arriving in their crisp frock and business suits, are dungaree-clad men and women tourists in T-shirts and no ties, sitting in their regular seats.

But Dr. Charles Trentham, the church's white-haired, Tennesseereared pastor who appears to relish his role as the President's minister, has cautioned his people: "To offer your seat to a visitor who may talk the rest of his life about this church and the President, this may be our contribution for four years and our mission, to sacrifice whatever comforts and intimacies we've known to hospitality."

For that reason, Dr. Sanks, who supervises staff operations, has refused to require tickets or invitations for worshippers, as some members have suggested. "That's not my idea of Christian worship," he said.

To beat the open-seating system, one longtime member says she "discovered" she needed a hearing aid - an "infirmity" that allows her to sit in a special row across the aisle from the Carters. The church staff close the chief executive's pew six rows from the front. It is, the staff claims, the best seat in the house.

Duties for Dr. Trentham and his staff have inadvertently changed, too. Tourists, indigents and job-seekers line up to greet Dr. Trentham on Sundays, asking him to introduce them to the President or appeal on their behalf.

Outsiders call Dr. Trentham on the telephone requesting jobs in the administration. He says he receives "several letters a day" seeking recommendations for ambassadorships, and recently, as head of the Peace Corps.

"A couple rode a bus from Boston to tell the President they had been evicted," Dr. Trentham recalled. "I said I was very sorry they've made the trip, but if they had asked me in advance I could have told them I have no access to the President.

On any weekday, up to two-dozen visitors arrive at the church's locked side door and state their business into an intercom. Few of them gained admission because, in the words of one church secretary, they are "kooks and strange people." They come begging for money or they want to give the President a gift, according to church employees.

Dr. Trentham said few gain admission. He said he does not like that, "but we can't turn them loose on the secretaries." If a church officer isn't available to assist, "we have to run a pretty hard-nosed operation," he said.

When alone, the congregation jokes about the confusion. "You know how you can tell a member from a stranger?" Ed Sonnenschein asked his fellow congregants at the church's 175th anniversary banquet at the Shoreham Americana in April. "The strangers wear trench coats, hearing aids and talk into their sleeves," he said, as the crowd guffawed at the swipe at the Secret Service.

While the church has experienced many new inconveniences since January, it has also enjoyed a growth in membership, the reverse experience of many other downtown churches that have lost members moving to the suburbs.

Since January, First Baptist's congregation has grown by 500 members, putting the total over 1,000, including the Carters and Attorney General Griffin Bell and his wife.

Two new members, Larry and Janet Manning, who moved here recently from California, said the President was an important factor in their decision to join.

"I heard him teach a Bible class, and you'd think you'd sit in awe just because it's the President," said Janet Manning. "That wasn't it. The awe for me was there in my Bible class was a man trying to live the nation's God-given ideals like the rest of us."