The Smithsonian Institution's persistence has paid off -- after 16 years of trying, it has won the right to take over the U.S. International Trade Commission Building and turn it into an American art archives and museum.

President Reagan signed legislation last month authorizing the transfer of the 140-year-old structure, sometimes called the Tariff Building, at no cost to the Smithsonian. It is still likely to be several years before traveling American art shows and other exhibits will be housed in the aging and deteriorating building at Seventh and E streets NW.

First, the General Services Administration, the federal government's landlord, must find a new downtown D.C. home for the trade commission, which has occupied the building for 63 years. Lorin L. Goodrich, the commission's director of administration, said he hopes a move will be accomplished by the fall of 1986, by which time the Smithsonian expects to have its renovation plans completed and the money to do the work.

The building transfer legislation authorizes $40 million for the project. Tom Peyton, the Smithsonian's director of facility services, said $3 million will be sought for the fiscal year starting next Oct. 1 to plan the renovation and $37 million the following year to do the work, all subject to the budget-tightening strictures of the Office of Management and Budget.

But after the seemingly endless struggle to win approval of the transfer, Peyton is pleased. "We're optimistic that GSA will find the trade commission a new home," he said.