Alexandria Mayor James P. Moran Jr. got a personal glimpse of the city's prostitution problem last month when he and City Manager Vola Lawson investigated a complaint about a brothel operating in the heart of the city. He was propositioned.

"A woman who was high obviously didn't know I was the mayor," Moran said recently, recalling the incident.

Tourists flocking to Old Town often discover that an age-old profession is thriving within a few blocks of the historic area.

Prostitution has plagued Alexandria off and on for more than a decade, according to police and city residents, who note that in recent years prostitutes have extended their business from side streets off King Street into nearby neighborhoods.

"I don't lose a lot of sleep over prostitution," Moran said. "It's not something we can ignore, but it's something we are keeping under control."

The level of prostitution in Alexandria does not begin to approach that of Logan Circle or 14th Street NW in the District, but law enforcement officers say Alexandria and other suburban areas are affected by the District's crackdown efforts.

"When D.C. chases them out of town, they move on to Alexandria, Arlington and Fairfax," one Alexandria police officer said.

Police describe the problem as a never-ending cycle: Prostitutes who are arrested are immediately replaced on the street by prostitutes who have just been released from jail.

"There is something of a revolving door . . . . They're always going to go back on the street," Moran said.

"People are constantly complaining that they don't want prostitutes and transvestites hanging around on their doorsteps, or cars parking in back alleys behind their houses," said one of the two undercover police officers charged with coordinating the battle against prostitution in Alexandria. "But chances are we will never completely eliminate it."

Police say that even with the AIDS scare and at least one Alexandria prostitute's death from acquired immune deficiency syndrome, business is flourishing.

Last week, an Alexandria Circuit Court judge closed a house on East Mount Ida Avenue -- evicting its tenants and prohibiting the landlord from renting it again -- after six people charged that "lewdness, assignation and prostitution is conducted, permitted and carried on" there.

Deputy Commonwealth's Attorney S. Randolph Sengel said that he did not have enough evidence to charge anyone with prostitution and that closing the house was the next best step. Sengel said that was the first action in the city under a 1982 state law designed to close down places where prostitution is alleged to be occurring.

Lt. Arthur L. Bratcher, who heads Alexandria's vice and narcotics unit, said the department made 80 prostitution-related arrests in 1985 and 41 arrests in 1986. As of last Wednesday, police had charged 16 prostitutes and customers this year. "Our problems are year-round," Bratcher said, "but it's worse during the warm-weather months."

Arlington police reported 23 prostitution-related arrests from July 1985 through June 1986, and nine arrests from July 1986 through June 1987. Fairfax County reported 20 prostitution-related arrests in 1985 and 10 in 1986; figures for 1987 were not available.

By comparison, District police made more than 1,565 prostitution-related arrests last year. Lt. Roger Jones, head of the prostitution branch of the D.C. police morals division, said the number of prostitutes traveling between the District and the suburbs is small. "Some of our girls might go there occasionally, and some of theirs do come into town," he said.

Alexandria's vice unit uses undercover officers and residents to pose as customers and prostitutes. Vice officers patrol the city after dark in unmarked cars and, if they lack sufficient evidence to press charges, they warn prostitutes and potential customers.

"We've seen positive results," said one of the two undercover officers spearheading the fight against prostitution.

Several officers said customers and prostitutes have been receiving stiffer sentences in recent years. "There was a time when the judges were very lax," one officer said. The maximum sentence for prostitutes in Virginia is a $1,000 fine and one year in jail, officials said; customers face a maximum $500 fine and six months in jail.

Alexandria police say their biggest problem is with transvestites, who investigators say outnumber female prostitutes 10 to 1 and often surprise unwary customers. "There are some up there that I would swear are girls," one undercover officer said.

Also of concern are thefts and assaults by prostitutes; police say these crimes often go unreported because the victims are married or in the military.

Merchants along King Street, Alexandria's main commercial strip, have been the most vocal critics of the prostitution problem, city officials said.

"It bothers me because it's in the same area as my store," said George R. Viteri, owner of Rauls of Old Town, a men's clothing store at 1017 King St. "People are afraid to walk in the area."

"I'd like to see something done about it because I'm sick of seeing it," said Paula L. Croisetiere, an employe of Freeman Advertising at 1112 King St.

Rose Williams Boyd, the city's citizens assistance officer, said that in the past year her office has received about half a dozen complaints from residents and merchants about prostitution. "The businesses have complained about the boldness of the prostitutes," she said. "Owners feel the prostitutes may be deterring . . . customers."

"There's a problem that everyone sees in the Old Town area with prostitutes and transvestites . . . ," the undercover police officer said. "Now we're receiving a lot more complaints from other areas of the city we didn't see before," including Del Ray and Arlandria, he said.

"In different areas, you bump into a few prostitutes," said Ted Karanikolas, owner of Tiffany Tavern, a restaurant and bar at 1116 King St. "They are here and there, but the problem is not like that in Washington."