Exterior security for foreign embassies has been the responsibility of the United States Secret Service Uniformed Division (previously called the White House Police Force and Executive Protective Service) since 1970 — though obviously, that mandate doesn't extend onto the embassy property.

Security within the embassy itself was the responsibility of the KGB. The KGB's First Chief Directorate, responsible for foreign intelligence, maintained a rezidentura (station) in every Soviet embassy, staffed with intelligence officers under 'legal' cover as Soviet diplomats (as distinct from the 'illegals'; intelligence officers who had no diplomatic cover and did not operate from the embassy.)

The First Chief Directorate was a vast service-within-a-service, with dozens of sub-departments. One of these, Line KR (for 'kontrazvietka'/'counterintelligence'), was responsible for internal security: within each overseas rezidentura, Line KR officers were tasked with maintaining the security of the embassy and monitoring all embassy personnel, to prevent either penetration by foreign intelligence services or defections by embassy staff.

Line KR's counterintelligence tactics were not strictly defensive — they also conducted operations against foreign intelligence services, to protect KGB operations and assets and identify enemy agents: arguably the two most important agents the KGB ever recruited inside the US, CIA officer Aldrich Ames and FBI agent Robert Hansson, were both recruited and handled by Victor Cherkashin, the head of the Line KR detachment at the Washington embassy.

Hansson, among other dubious honours, was responsible for alerting the KGB to an ambitious attempt by the FBI and NSA to bug the new Soviet embassy then under construction in Washington, by digging a tunnel under the embassy and packing it with surveillance equipment.

If you're interesting in reading more about Soviet intelligence, check out Christopher Andrew's The Sword and the Shield and The World Was Going Our Way, Jonathan Haslam's Near and Distant Neighbours, and Allen Weinstein's The Haunted Wood.