Mario M. Cuomo, saying he was embarking on "a fierce test of our resolve," became New York's 52nd governor yesterday. Michigan and New Mexico also inaugurated new governors.

Cuomo, succeeding fellow Democrat Hugh Carey, said in his inaugural address, "I believe government's basic purpose is to allow those blessed with talent to go as far as they can. But I believe that government also has an obligation to assist those who, for whatever inscrutable reason, have been left out by fate--the homeless, the infirm, the destitute."

In Lansing, Mich., former congressman James Blanchard declared unemployment--17.2 percent in Michigan in November with more than 700,000 jobless--to be the state's No. 1 enemy.

"We must all work together so that all can find work," Blanchard said. "It is joblessness that is our common enemy. And it is against this common enemy that we must make a common cause."

Blanchard, a Democrat, succeeds Republican William Milliken.

Toney Anaya became New Mexico's 26th governor in Santa Fe. He said the state could show the world that "cultural diversity can indeed form the basis of mutual understanding."

"We enter 1983 with staggering problems facing us, and the enormity of those problems demands that those of us elected to office place our partisan and petty differences aside. If we do not, we will fail the tests ahead."

Anaya, a former state attorney general, succeeds fellow Democrat Bruce King.