SEATTLE Mayor Norman Rice wants his town to be an "international city."Little wonder, then, that he is a great believer in expanding overseas trade and valuing local diversity - welcoming the thousands of newcomers who have settled here in recent years from all over the globe. From the Rainier Valley in southeast Seattle to Aurora Avenue on the north, dramatic change is evident for this cosmopolitan community once dominated by Northern Europeans - from England, Scandinavia, and Germany; now store signs greet visitors in Vietnamese, Korean, Chinese, Cambodian, Russian, and Spanish. As Mayor Rice notes, Seattle has always looked outward: Sailing ships carried lumber and wheat from the Pacific Northwest to the world; now Boeing jets and Microsoft computer products are carried overseas. But if Seattle is, as Rice says, an international city that he would like to see become even more international, it is also a typical urban city in 1990s North America. That means that the mayor, a youthful-looking, energetic and genial individual, spends most of his time on such basics as finding the spare financial resources to meet pressing budget deadlines, curb crime and drug use, seek greater affordable housing, and ensure that local schools fulfill student needs. Indeed, listening to Rice comment on his challenges, a visitor is immediately reminded of almost exact woes faced by the mayors of such cities as New York and Chicago. Case in point: Rice just closed some downturn parks at night to block drug distribution; New York Mayor David Dinkins has had to do the same thing. And gangs? They don't belong just to Los Angeles. Mayor Rice has expanded a city antigang program. Rice - who is called "Norm" here - has been scrambling to meet a projected $24 million shortfall in the city's $370 million budget for fiscal year 1992, beginning next Jan 1. To avoid new taxes, the city is seeking to cut personnel and reorganize a number of agencies. Mayor Rice, born in Denver, moved to Seattle in 1968 to attend college. After later serving for 11 years on the city council, he was elected mayor in November, 1989, with 57 percent of the vote. Because he was the first African-American elected mayor in a city that is 75 percent white, he drew widespread national press attention. Midstream into his term, he continues to be very popular. Rice says he assumes he will not be able to finish all that he would like to do in his first term - which means that he presumably will run for a second term that would begin in January 1994. Rice has had his hands full on the crime front lately. Some city council members have faulted the police department for an inadequate job in controlling drug-dealing in downturn areas, such as near the Pike Place Market, a major landmark. Besides closing a number of parks at night, the city is making it harder for repeat drug dealers to get back on the streets. Mayor Rice believes that good schools, adequate health care, and jobs are as important as police work in controlling crime. "When the economy tightens up, the level of violence goes up," he says. But such social programs can be costly. Are United States cities getting enough help, either from Washington, D.C., or their state governments? Mayor Rice shakes his head in the negative. Although noting that the states have financial problems of their own, he faults Washington for what he feels is its indifference to American cities. This has not been a totally upbeat period for Rice; baseball commissioner Fay Vincent came to Seattle and raised the possibility that the Seattle Mariners baseball team might be sold and moved elsewhere. Late last week it was confirmed that the team might, indeed, have to move, even though 1.5 million fans have bought tickets this year. Still, nitty-gritty public issues aside, Rice delights in the diversity of Seattle, although he wishes "the top management" of corporate front offices were a little more open to minorities and women. While whites are most numerous here, Asian-Americans now constitute the second largest grouping, at around 9.7 percent; African-Americans make up about 9.5 percent; Hispanics about 3 percent; Native Americans about 1 percent. Seattle, says Mayor Rice, must be so international in make-up that if a foreign bus inessman arrived here, he would immediately feel right at home. That's the only way US cities will be able to globally compete in the future, he says.