What was going on when I was born?

New York City's $16.7 billion budget for the next fiscal year was adopted yesterday after three days and two nights of negotiations that produced more money for public schools, police officers, street cleaners, parks and libraries. The approvals by the City Council and the Board of Estimate included an increase in the property-tax rate that will average about 9 cents, bringing the average rate to $9.21 for each $100 of assessed value. Charges for water and sewer use are also to go up, by an average of $9 annually, in the fiscal year 1984, which begins July 1. The plan must now be approved by the State Financial Control Board, the city's chief fiscal monitor. The adoptions, carried out in a brief, good-natured manner, ended deliberations that had left virtually the entire top rank of municipal government red-eyed and irritable. The Board of Estimate vote was unanimous, while the Council had no more than three negative votes, of 42 cast, on the four separate budget bills it approved.

The State Department today welcomed an appeal from Yuri V. Andropov for improved relations and asserted that if Moscow was ready ''to take concrete steps'' it would ''find a ready partner'' in the Reagan Administration. ''For our part, the U.S. approach to relations with the Soviet Union is serious and seeks to explore realistic ways of cooperation to mutual advantage,'' Alan D. Romberg, a State Department spokesman, said. ''Our diplomatic dialogue with the Soviet Union on all outstanding issues is intensive and comprehensive, and will continue.'' On Thursday Mr. Andropov, talking with W. Averell Harriman, a former United States Ambassador to Moscow and a longtime specialist on Soviet affairs, said the Soviet leadership was ''ready and interested in seeking joint initiatives'' that would improve relations with Washington.

Murder charges against a 29-year-old Brooklyn man, accused of killing his mother by mailing her an explosive device in a hollowedout cookbook, were dropped yesterday. The charges against the man, Craig Kipp, were dismissed in State Supreme Court after the Brooklyn District Attorney's Office said the evidence was not sufficient to prove his guilt in the death in May 1982 of his 54-year-old mother, Joan, at her home in the Bay Ridge section of Brooklyn. Police officials declined to say whether there were other suspects in the case, but a Police Department spokesman, Deputy Inspector Robert Burke, said that ''the investigation will have to resume'' in the death of Mrs. Kipp, who was a supervisor of guidance counselors in the city school system. Remains of Device Tested The key evidence against her son, according to an assistant district attorney, Charles Abercrombie, included conclusions about the work of a dog trained in tracking human scents and the finding by a handwriting expert that samples of Mr. Kipp's printing were similar to the printing in the message accompanying the device.

International The State Department welcomed an appeal by Yuri V. Andropov for improved relations. The State Department said that if Moscow was ready ''to take concrete steps'' it would ''find a ready partner'' in the Reagan Administration. The Soviet leader made his appeal at a meeting Thursday in Moscow with W. Averell Harriman. (Page 1, Column 6.) Unease is growing in Argentina over a series of threats, beatings and, in some cases, killings, by politically extremist underground groups. An abiding fear in the country is that the violence might undermine national elections scheduled for October by the military Government. (1:5.)

Gretchen Nicol, a nurse-practitioner who completed advanced studies in a federally financed program at Ohio State University, has examined hundreds of women for early-warning signs of cancer, working in clinics, a doctor's office and as an Air Force Reserve officer. Last summer at Health One, a Blue Cross-affiliated clinic in a middle-class neighborhood in Columbus, Miss Nicol gave a routine checkup to Linda Skaggs, an employee of the Ohio State Medical Board. The board thereupon accused Miss Nicol of ''illegal practice of medicine'' and sued her and Health One in Franklin County Common Pleas Court, seeking an injunction to prohibit examinations for cancer signs by nurses. The case is but one in which the medical authorities in more than a dozen states are using legal and economic weapons to limit the scope of health care administered by 20,000 registered nurse-practitioners. The Ohio medical board, which is run by and regulates physicians, contends that ''breast and pelvic examinations, with an associated Papanicolaou smear,'' or ''Pap smear,'' are ''prohibited acts of medical diagnosis.''

Extremist underground groups have been stirring unease here in recent months by resorting to a growing number of threats, beatings and, in some cases, killings. An abiding fear is that the emerging violence may undermine national elections scheduled for October by the military Government of President Reynaldo B. Bignone. The military, after seven years in power, has promised to step down in January. The violence remains limited; this capital has only a normal police presence in the streets. But the violence appears directed at specific targets.

The nation's unemployment rate fell slightly in May to 10 percent from 10.1 percent as employment increased and the size of the labor force was unchanged, the Labor Department reported today. Both the seasonally adjusted unemployment rate and the total number unemployed, the 11,192,000 Americans out of work and looking for work, remained at one of the highest levels since 1940. But May was the third successive month in which the unemployment rate declined, and the Commissioner of Labor Statistics, Janet L. Norwood, said there were marked signs of further improvement in the labor market. ''The improvement in the automobile industry is particularly noteworthy,'' said Dr. Norwood. ''Since November, employment in this industry has increased by 105,000 and the unemployment rate for automobile workers, at 14.3 percent in May, was more than 10 percentage points below its November recession peak of 24.9 percent.''

New Jersey officials yesterday identified a second site that may be contaminated with dioxin and said nine chemical plants now in operation would also be investigated for the toxic compound. The Commissioner of the State Department of Environmental Protection, Robert E. Hughey, said guards and the police had closed off the abandoned two-acre area in the Middlesex County community of Edison Township, which was the site of the former Insecticide Corporation. He said tests would begin next Tuesday. The announcement, at a news conference in Governor Kean's office in Trenton, came as inspectors from the Federal Environmental Protection Agency searched a Newark neighborhood where high levels of dioxin were found in soil samples at the closed Diamond Alkali Company chemical plant.

''Speak Russian, please,'' the ticket agent said, turning away from the swarthy traveler standing at his window. The traveler, unable to make himself understood in his native Kazakh, ceded his place at the head of the line and walked away disconsolately. The scene was the railway station at Alma-Ata, capital of Kazakhstan, one of the Soviet Union's 15 republics. But it could have been in a number of regions where language has become the focal point of wider strains between ethnic Russians, who dominate the country, and citizens of other nationalities, who make up nearly 50 percent of the 270 million people in the Soviet Union.

Chancellor Helmut Kohl met today with Paul H. Nitze, the American negotiator at the Geneva talks on medium-range missiles, and later said through a spokesman that he believed the United States was seriously and responsibly seeking an accord with the Soviet Union. As the opposition Social Democratic Party sharpened its criticisms of the plans to begin stationing American missiles here in December in the absence of an American-Soviet accord, it was announced that Mr. Nitze would hold another session with the Chancellor on June 30, four days before Mr. Kohl travels to Moscow. Mr. Nitze's intensive consultations with the Kohl Government are intended both to keep the Chancellor abreast of the details of the Geneva discussions and to demonstrate to an occasionally skeptical West German public that the Reagan Administration is interested in reaching an arms limitation accord with the Soviet Union.

At 7:50 this morning Keith Best, the Member of Parliament for Ynys Mon, was sitting in shirt-sleeves at the kitchen table in his modest row house at Holyhead in northwestern Wales, painstakingly cutting out newspaper articles about himself for reproduction in his final campaign broadside. By 9:30 he was standing in the rain in a pinstripe suit outside an apartment in a public housing project in Cemaes Bay, overlooking the windswept Irish Sea, listening patiently while a constituent complained about the delays she was experiencing in getting into a National Health Service hospital for the hysterectomy her doctor had told her she needed two months ago. Tonight he spoke at one of the eight public meetings of his campaign for re-election, a gathering of about 100 people in the school hall in the little resort community of Benllech. To a degree that would astonish most Americans, accustomed as they are to lavishly funded campaigns, British general elections are small-scale, do-it-yourself affairs. Votes are sought retail, not wholesale.

A report published today by Tass, the Soviet press agency, said Mongolia had forcibly repatriated some Chinese residents to China. The report, based on an account in a Mongolian newspaper, Hodolmor, denied a Chinese Foreign Ministry statement that Mongolia had begun a widespread policy of expelling Chinese. Instead, the Tass account said, the expulsion was being used only against ''individual Chinese citizens who systematically and maliciously violated'' Mongolian law. At the same time, Tass repeated the Mongolian newspaper's charge that the overwhelming majority of Chinese in Mongolia ''shirk socially useful work,'' a characterization that was once common in statements about China but that has disappeared as Moscow has tried to improve relations with Peking.