For 27 years, the Putt-Putt Golf Course in the Ballston area of Arlington has been a community fixture. As late as 11 p.m. on humid summer evenings, the place is packed with couples, families and friends, including more than a few teen-agers who say there are few other places for them to go.

But the Putt-Putt's days are numbered, too, unless county officials can find another location for the amusement facility.

Construction of a high-rise office and residential complex is scheduled to begin early next year near the miniature golf site, on Wilson Boulevard near Glebe Road. The Putt-Putt, which is open from April through September, will be given a month-to-month lease until the wrecking ball comes calling, according to Peggy Dubynin, manager for the high-rise project, called The Ellipse.

County officials, eager to preserve that scarcest of civic resources -- a recreational facility that appeals to grandparents as well as teen-agers and tykes -- are looking for a new location for the Putt-Putt franchise or something similar.

John G. Milliken, vice chairman of the Arlington County Board, has taken a personal interest in the search.

"I really want to find a new place for it," said Milliken, who has been a patron of the 36-hole course since it opened in 1960. "I've seen . . . three generations all playing at the same time. I'm looking for that kind of recreation opportunity for people."

Milliken said his concern also was spurred by the lack of organized activities for the area's teen-agers. "There are very few things a 14-, 15-, 16-year-old can do," he said. "They can go to the movies, wander around the shopping mall."

Many of the young people at the Putt-Putt on a recent night agreed.

"It's just something to do," said Kirk A. Baker, who was celebrating his 20th birthday with several friends.

"You have to be 21 around here to go dancing or anything," added Shah Shahbazi, 18.

As Arlington's population ages -- only 17 percent of it is under 18, according to 1980 census data -- the demand for recreational facilities has lessened, Milliken said.

Redevelopment has also taken a