Prosecution of young men who have failed to register for the draft will not begin until March under the Reagan administration plan to give non-registrants a "grace period" in which to comply with the law, a Selective Service System spokesman confirmed last week.

The government will probably continue to accept late registrations even after March 1, but the Justice Department has not yet made any final plans for its prosecution program, Joan Lamb, chief Selective Service spokesman, said.

Public Relations Drive

In the meantime, the draft agency has begun a nationwide public relations campaign to persuade the estimated 800,000 young men who have failed to register to do so.

Selective Service plans to ask Congress for a $400,000 supplementary appropriation to add to the $150,000 already set aside to conduct the campaign, Lamb said.

Selective Service hopes to use the money to buy radio and television spots and to remind young men by mail of the President's recent decision to continue registration, she said.

In addition, the agency will send letters to college newspapers and high schools, asking their help in publicizing the draft registration law.

The government estimates tht 6.6 million young men have registered since July 1980, when President Jimmy Carter resumed the sign-up process that President Gerald Ford suspended in 1975.

No Convictions to Date

Non-registrants are subject to a maximum penalty of five years in jail and a $10,000 fine, but the Justice Department has not yet brought any cases before a grand jury.

In fact, Justice has the names of only about 150 registration resisters provided by public informants. Selective Service, the agency responsible for identifying non-registrants, has not begun an active program to track down others who have broken the law.

On Saturday, about 100 local anti-war activists protested Reagan's decision to abandon his opposition to registration and continue the program by marching on Washington St. in downtown Boston.