What was the American response to the Sino-Soviet border conflict?

Yes, the response was called "Nixon goes to China."

(Republican) President Nixon was a "cold warrior," but he was always looking for a "reversal of alliances.'

On the domestic front, he was hoping that the Republicans could "crack" and ultimately win the "solid" (and formerly Democratic) South, following Lyndon B. Johnson's institution of the Civil Rights Act of 1964. (His vision was eventually realized in a "Republican" South).

The most troubling foreign policy issue of him (and Eisenhower) was the "two on one" aspect of the Soviet Union and China versus the U.S. Nixon saw a chance to "split" that (and win) China over to the U.S. side.

His visit to China resulted in a "normalization" of relations with China that led to an end to the Vietnam war, a reduction of tension over Taiwan, the diplomatic "isolation," of the Soviet Union, and above all, the opening of trade relationships that led to the U.S. and China becoming important trade partners to this day.

Common view certainly is that the US supported China.

The mainstream press has just come out with a Chinese expose that has informed an undoubtedly surprised world that in 1969 the USSR wished to settle its historical score with China and launch a nuclear attack. The USSR merely wanted an assurance of U.S. neutrality. Far from the USA welcoming this de-clawing of the growing dragon, it instead threatened that there would be retaliation from the U.S. against Russia.

According to a report first carried in the Daily Telegraph, Chinese historian Liu Chenshan, writing in an officially sanctioned newspaper, stated that the threat “came in 1969 at the height of a bitter border dispute between Moscow and Beijing that left more than one thousand people dead on both sides.”

Liu Chenshan further noted:

On 15 October 1969, he quotes Soviet premier Alexei Kosygin as telling Soviet leader Leonid Brezhnev that Washington has drawn up "detailed plans" for a nuclear war against the USSR if it attacked China.

"(The United States) has clearly indicated that China's interests are closely related to theirs and they have mapped out detailed plans for nuclear war against us," Kosygin is said to have told Brezhnev.

That same day he says Anatoly Dobrynin, the Soviet ambassador to Washington, told Brezhnev something similar after consultations with US diplomats. "If China suffers a nuclear attack, they (the Americans) will deem it as the start of the third world war," Dobrynin said. "The Americans have betrayed us."

It also provided a chance to the US to open up to China, when they increasingly considered the Soviets to be the greater threat.

Archival documents also illustrate secret White House initiatives during the summer and fall of 1969 to turn a page in Sino-American relations. Convinced that Sino-Soviet tensions provided a basis for rapprochement but also determined to minimize the State Department's role, Nixon and Kissinger tried to open secret communications with China through Pakistan and Romania. Other documents show how State Department officials tried to assert a role in policymaking on rapprochement and, before they were cut out altogether, made important contributions to White House efforts to signal a friendly interest in communications with China.

From the birth of the People’s Republic of China in 1949, both China and the United States wanted to improve their relationship. Ideological differences and mutual suspicions prevented a common understanding. So when the Sino-soviet conflict happened, it was a chance for the US to patch up.

Naturally the Nixon administration took advantage of this open confrontation; it perfectly fitted Nixon’s plan to normalize relations with China. It was hoped, though repeatedly denied, that the ‘China card’ would be useful to obtain concessions from the Soviet Union in forthcoming arms reduction negotiations. Source- Sino-American-Relations

The Counter View:

Patrick Tyler says in his controversial book that the Nixon administration was willing to accept a Soviet attack on China in order to secure Moscow's active help on a Vietnam war settlement.

Three days after Kissinger's meeting with Dobrynin, Nixon, in a news conference, portrayed China as a global nuclear menace of equal concern to both the United States and the Soviet Union. In deciding to press forward with developing a limited antiballistic missile system to safeguard the continental United States* in the event of a nuclear attack, Nixon emphasized just how dangerous China had become to both superpowers.

"I would imagine that the Soviet Union would be just as reluctant as we would be to leave their country naked against a potential Chinese Communist threat. So the abandoning of the entire [ABM] system, particularly as long as the Chinese threat is there, I think neither country would look upon with much favor."