What was the program of the Mensheviks in their later years? How did they hope to achieve it?

It is a long story. I am not completely sure which period you are actually interested in. Mensheviks (RSDRP) ceased to exist as a party in 1951, so I will address the earlier period, up to the 2nd World War. RSDRP in immigration was split in the left and right wing. Roughly speaking, the left (Dan, Gurevich, Schwarz, et. al) following "Martov Line", were hoping to find a common ground with the "moderate" bolsheviks (like Bukharin), they were also hoping for an evolution of the USSR into something more resembling socialism (as they understood it). They were also considering the economic developments in the USSR as the developments of the "state capitalism", which was supposed to, eventually, lead to true socialism. The right wing (Potresov, Gravi, Aronson, et al) , in contrast, were hoping for a popular uprising in the USSR (once things turn really bad - and they did, especially during collectivization). The right wing of RSDRP regarded Soviet system as a form of fascism and did not expect its evolution into a an actual socialism. By the end of the 1938 (after Moscow show processes, mass terror, etc.), and especially in 1939 (after Molotov-Ribbentrop pact) both wings realized that they were wrong. RSDRP in immigration understood very well that their underground organizations in the USSR were completely wiped out. Thus, by publishing in the "Socialist Messenger", Mensheviks were primarily talking to each other (and their numbers were tiny), it was a way to flash out their ideas and disagreements. Their only semi-realistic way to effect the events was through Socialist International (RSDRP was part of it). However, due to their internal split, Mensheviks could not accomplish even that. I am not sure what RSDRP position during the WWII was, if I had to guess, they were again split on how to support the Soviet Union in this situation (i.e. support 100% or support, but to condemn the Soviet totalitarian system at the same time).

Incidentally, after the WWII, some of the Mensheviks, e.g., Dallin and Nikolaevsky also known as Nicolaevsky, again became relevant. Among other things, they interviewed Soviet DPs who found themselves in the Western Europe, to record the plight of the ordinary Soviet citizens under Stalin's rule (D.Dallin, B.Nicolaevsky, "Forced Labor in Soviet Russia", YUP, 1947). Since Mensheviks were acutely interested in what was happening in the USSR, they turned out to be useful in the beginning of the Cold War. For instance, once the Russian program of Radio Liberty was established in 1950s, it would invite Mensheviks for their broadcasts (rather than, say, monarchists), I can explain why if this is interesting. The same Nicolaevsky (who immigrated to the US in 1940) was one of the first American "sovietologists" and is regarded by some as "the father of Kremlinology" (see "The Anti-Communist Manifestos: Four Books That Shaped the Cold War", by John V. Fleming, 2009).

There are several sources you may want to take a look at, for instance, "From the Other Shore: Russian Social Democracy After 1921", by André Liebich, HUP, 1997.

Most of my sources are in Russian, so I am not sure if you will find them useful.

There was another party you may want to read about, the Socialist-Revolutionary Party, more precisely, its right wing. In exile, they continue to function through 1940s.

The Mensheviks were "moderate" socialists. They favored a form of "creeping," even Fabian Socialism. For instance, they thought that the socialist movement should be led by the "middle" class against the capitalists. Many of them felt that Russia was the wrong place to begin the socialist revolution because it had such a small middle class.

The Bolsheviks wanted an "immediate" revolution. For them, a working class base was fine, as was a Russian locale. They felt that workers were violent enough to bring about the revolution, no need for the "peaceful" methods of the Mensheviks, who wanted the workers to benefit from, not lead the revolution.

The later Mensheviks split into two groups. One group, under Fyodor Dan, who died in the 1940s, felt that the Bolsheviks were more nearly correct, even while regretting their Stalinist excesses of the 1930s, and the later murder of Trotsky. His following disappeared by about 1950, as noted by another poster. A much larger group under Raphael Abramovich, who moved to New York City and maintained the Menshevik Center there, as noted by the other poster. They maintained the movement until the mid-1960s and felt that the Bolsheviks and their excesses were the worst thing that could have happened to socialism. They wanted to "start over" and ironically felt that perhaps the best check on Bolshevism was capitalism. This view proved to be prophetic when Ronald Reagan's 1980s defense build up brought down the soviet Union in the early 1990s.

edit by OP: previous version of this question was whether the Mensheviks still controlled their paper, Socialist Messenger, into the 1960s. Having established they still existed until at least 1965, I changed the question.

I'm not an expert, but it seems that the Socialist Messenger was first published by the Menshevik Centre in Berlin in the early 1920s. It moved from Berlin to Paris with the Menshevik Centre in 1933, and again from Paris to New York in 1939.

Publication seems to have remained with the Menshevik Centre until it closed in either 1965 or the early 1970s (depending on which source you believe). I can't find any source that suggests that the Menshevik Centre was taken over by any other faction, so it seems that the Socialist Messenger was still controlled by the Menshevik movement until the end.