Attached are the endorsement editorials for Willkie, Dewey, and Eisenhower.
Willkie was endorsed for four reasons: his greater commitment to war readiness, his support for existing government programs and belief in what the Times calls "increased production" (meaning Willkie preferred economically outgrowing the Great Depression rather than Roosevelt's inclination towards wealth redistribution), Roosevelt's support of the now-widely accepted Keynesian economics (which they dismiss as a "politician's paradise"), and concerns for breaking the two term precedent.
Dewey was endorsed in 1948 because of the Republican Party's shift away from isolationism, and his much greater experience in foreign policy compared to his 1944 presidential campaign. The Times also felt that the Democrats were no longer able to effectively govern, due to the three-way split in the party in 1948, and Truman's indecisive policy-making.
Eisenhower was endorsed in 1952 due to the Times' continued disagreements with Truman's policies, Stevenson's vagueness on certain issues, and the belief that as a former general, Eisenhower was positioned to rebalance the economy away from the military footing that had continued after the Second World War. They made mostly the same arguments in 1956, and believed that Eisenhower's record was good enough to warrant re-election.
In short, The New York Times has always been a centre-left (emphasis on centre) internationalist newspaper. As the Republican Party moved towards a more internationalist stance in the 1940s, and began to accept at least certain parts of the New Deal, the GOP's presidential candidates became more palatable to them. As the Republican Party began to drift back to the right with the nomination of Nixon in 1960, the Times returned to their Democratic preference.
http://graphics8.nytimes.com/packages/pdf/opinion/timeline/willkie-1940.pdf
http://graphics8.nytimes.com/packages/pdf/opinion/timeline/dewey-1948.pdf
http://graphics8.nytimes.com/packages/pdf/opinion/timeline/eisenhower-1952.pdf
http://graphics8.nytimes.com/packages/pdf/opinion/timeline/1956eisenhowerpage.pdf