Bearing letters of protest and signs that said "Don't Sock It to The Shoe," about 20 windblown, well-bundled youngsters from The Shoe after school care center in upper Northwest and nine of their mothers demonstrated in front of the District Building yesterday to protest the city's plan to close the center.

The protest apparently worked, since before the excited, red-cheeked youngsters left the District Building, they had promises of help from Mayor Barry, James Gibson, director of Planning and Development, and Ward 3 council member Polly Shackleton.

The 3-year-old center was scheduled to close yesterday after the Board of Zoning Adjustment denied a request from the center's operator, Judith Heintz, for a zoning exception to operate the center at her home in her expensive residential neighborhood near Chevy Chase Circle.

Gibson, who said his office had recommended the zoning board grant the exception, encouraged Heintz to ask the board to reconsider her case. Shackleton and Warren Graves, a Barry assistant who spoke for the mayor, meanwhile, promised to use their influence to keep the service open.

"There are not enough day care centers in the city and we are encouraging people to open them. So it is incumbent on us to keep the ones we have open if they are not in flagrant violation of the health code or other city laws," Graves said.

Under city laws, only child care centers that operate all day need a special license from the Department of Human Services. Before and after-school programs, such as Heintz', do require a certificate of occupancy from the Department of Licensing, Inspections and Investigations. Before Heintz could get the certificate, she had to get the zoning variance, what has proved in past cases to be a fairly routine matter.

But in the Heintz case, one neighbor, complaining of the children's noise, found a 1925 covenant on the Heintz house which specifically prohibits the use of the house for a business. The zoning board ruled against Heintz in large part because of this covenant.

Though the youngsters and their mothers may have won one round yesterday, the fate of The Shoe is still uncertain. If Heintz continues to operate until her appeal is set, she could face a fine of $100 a day. Her case has been referred by the Licensing Department to the Corporation Counsel for prosecution.