February 18, 1982

BOSTON SITTING with Nina Fletcher Little is a bit like sinking gently into history. One discovers a young America. The mullioned windows of her sitting room, framed when the Massachusetts farmhouse that is her principal home was built in 1812, look out on snowy woods, gray and white in the dusk of winter. The tall clock by the door chimes softly and then ticks on, as steadily as it has since her great-greatgrandfather, a Salem clockmaker, built it around 1820. Thomas Jefferson would have approved of her desk. A fall-front model of shiny brasses stamped with the third President's profile, it was made in 1790, 36 years before he died, and was passed down to Mrs. Little by a cousin. Some of the lamps and glassware would have seemed old to Thomas Alva Edison. Alexander Graham Bell would have recognized the telephone, which was installed in 1921, the year before he died.

Governor Carey proposed today a broad program to change the way public education is financed in the state. It would include a one-cent increase in the state sales tax, now 4 cents on the dollar, and would call for $8.6 billion in new state education aid over the next five years. If approved by the Legislature, the increase would raise the total sales tax to 9.25 percent in New York City and Yonkers and 8.25 percent in Nassau, Suffolk and Westchester Counties. Its impact would be partly offset through a new income tax rebate to low-and middle-income residents that would eventually return $300 million a year to New Yorkers whose income is below $25,000 a year.

''Europe is in greater danger than the Americans have understood so far.

The executive director of New York State's Financial Control Board warned yesterday that New York City's finances were in their most perilous shape since the crisis of the mid-1970's. In addition, a report from the board's staff said, there is ''an unusual degree of uncertainty'' in city plans to balance the budget for the coming fiscal year. ''The city has not had to face challenges of such magnitude in half a decade,'' said Comer S. Coppie, executive director of the Control Board. As the principal fiscal watchdog of the city, the board must approve all budget proposals and contracts.

David A. Stockman, the budget director, indicated some flexibility in the Administration's plans for cutting the budget. Testifying under heavy criticism before the House Budget Committee, he said that some portions of last year's tax cuts could be scaled back or repealed to increase Government revenues. (Page A1.) Industrial production fell 3 percent in January, the largest decline since May 1980, the Federal Reserve Board reported. The decline resulted from industry's efforts to work off inventories in the face of a worsening economy and poorer sales, as well as bad weather. (D1.) Housing starts fell 0.6 percent in January, after rising slightly in December, the Commerce Department reported. (D1.)