Visitors to New Mexico may still ask whether they need visas and immunizations, and the Treasury Department may need a refresher course in geography. But the U.S. Senate, which knows where all its 50 states are, has reaffirmed what New Mexicans have taken for granted for nearly 75 years: Their state is a part of the Union.

The Senate approved a resolution to that effect Friday in response to a letter from Treasury informing David F. Cargo, a former governor, that interest on a Treasury note was being withheld because Cargo's address indicated he was "not a citizen of the United States but a citizen of New Mexico."

An exchange of letters between Cargo and the department was printed Thursday by The Washington Post and caught the eye of Sen. Pete V. Domenici (R-N.M.), who got the Senate to approve a resolution designating Friday as New-Mexico-Is-a-State Day.

It was not the first time New Mexico had been confused with plain old Mexico, Domenici told his colleagues. His staffers often are transferred to the foreign affairs desk when they call the State Department, he noted, and residents who try to use New Mexico driver's licences to cash checks in the nation's capital are often refused because the licenses are assumed to have been issued by a foreign country. Moreover, visitors to New Mexico often ask about visas, immunizations and the "drinkability of our water," he complained.

When Domenici finished his lament, Sen. Warren B. Rudman (R-N.H.) asked whether New Mexico might have been eliminated by the Gramm-Rudman-Hollings budget-cutting law.

In any case, Rudman suggested that Domenici reconsider the whole issue. As a foreign country, he explained, New Mexico could qualify for foreign aid. A Hardy Welcome . . .

The Senate Finance Committee recommended last week that the Senate confirm Dorcas Hardy as commissioner of the Social Security Administration, sending the question to the floor on a unanimous voice vote.

No serious opposition is expected on the Senate floor, although Sen. Daniel Patrick Moynihan (D-N.Y.) asked that a vote be delayed until Hardy explains how she would comply with the Supreme Court decision last week that required Social Security to review the cases of some 10,000 mentally disabled New Yorkers who were cut from the rolls between 1978 and 1983. Aribusiness . . . If the Senate confirms her, Kathleen W. Lawrence soon will become the highest-ranking woman at the Agriculture Department. Reagan has chosen Lawrence to succeed Frank W. Naylor Jr. as undersecretary for small community and rural development, which includes the farm and housing loan programs of the Farmers Home Administration (FmHA).