Great question - thanks to /u/Georgy_K_Zhukov for bringing it to my attention!

To contextualize and work through Science in History requires a bit of understanding of what the "history of science" looked like in 1954, what sympathies and leanings would have been held by historians, and how you gauge the credibility of something compared to its importance. There are plenty of very important works that have significantly altered disciplines, or spurred new ones, that we would not now view as credible. In fact, nearly all foundational works have been revisited, altered, or even rejected entirely (one myth that Bernal perpetuates, for example, is that it was widely believed the Earth was flat in the Middle Ages). We might reasonably expect that such a capacious goal as the history of science is unlikely to be achieved - there are many such examples of overstepping in Bernal, but that doesn’t necessarily nullify the rest of the collection.

Bernal’s broader arguments are somewhat questionable, but not because of these generalizations or errata, rather the theory he is reliant on.

The history of science is a rather young field; the physician Henry Smith Williams and chemist George Sarton are the central figures of the early discipline. Einstein and Infeld write an interesting history of physics (interestingly, edited by CP Snow, who is now chiefly remembered for his Rede lecture “The Two Cultures,” which captures some of the problems of the discipline [a]). It is in the 1950s and 1960s that we get collections by Koestler, Asimov, Agassi, Neugebauer, and Kuhn that reframe the field, much like Bernal was working toward, but some of these projects are more successful than others for a variety of reasons. Bernal is one of these early figures, and is very important to the field - an analogous case might be Eliade, who isn’t necessarily credible but remains foundational.

At the same time we have these changes in the history of science, we have a very different impression of the Soviet Union than we might presume. Against the backdrop of the Cold War, many prominent intellectuals, chiefly French and British, were enamored with the promise of the Soviet Union, largely following a tradition from the 1930s of Soviet apologetic travelogues. After 1956 and 1957 many of these scholars would turn away from the USSR and by the 1980s would denounce it, but not all (e.g, Hobsbawm). I strongly recommend Francois Furet’s The Passing of an Illusion as an introduction to Western fascination with the Soviet Union; Andre Gide’s Return from the USSR is an excellent work as well. Bernal’s The Social Function of Science fits right into this time, and is valuable in critically examining science in a way that we don’t return to until the 1960s and 1970s, but makes many specious claims about the Soviet system. Hollander has some excellent discussions on intellectualism and Communism that will touch on this thread.

In “Engels and Science, a brief pamphlet by Bernal, he outlines some of his views on Engels and Marx as scientists.

”The crises of modern science appear in the first place intellectual difficulties arising from new and apparently incompatible discoveries. The resolution of these crises, that is, the process of bringing them into harmony with the general movement of human thought and action, is a task for the Marxist scientists of today and tomorrow...We have through dialectical materialism a greater comprehension of whole processes, which before were only seen in their parts.”

He also makes some very curious claims about Naturphilosophie that, while noble in the effort to examine legitimately Hegel and Schelling, probably extend a bit too far. Chiefly this brief pamphlet is an excellent introduction to Bernal’s approach: dialectical materialism.

Andrew Brown’s biography of Bernal is worth taking a look at for a much richer discussion of his life; of his views of the Soviet Union in particular, Brown offers that

”At the time Bernal wrote ‘Unholy alliance’, Stalin was ordering the liquidation of the kulaks, or smallholders, and the collectivization of farms. The utter desolation that soon followed dwarfed any biblical famine, yet was blithely denied by visiting Western intellectuals like George Bernard Shaw and the Webbs. That Bernal felt able to publish this essay in a collection nearly twenty years later, with no apparent embarrassment...showed that he had still not tumbled to the big lie. Bernal always insisted on portraying Marx as a scientist, but in assessing Marxism he failed to heed Rutherford’s wise dictum about the dangers of endless theorizing in the absence of supporting evidence.” (77)

and

To [Norman] Pirie, there seemed to be a religiose quality to Bernal’s belief in the Soviet system (111)

and

Like many European communists in the early thirties, Bernal thought that the Soviet Union represented the closest to an ideal, progressive society, and that the power of capitalism would soon be broken in Western Europe. (124)

We can’t entirely fault Bernal for still crediting the aspirations of the Soviet Union after it was clear they wouldn’t be achieved; many other intellectuals did the same. It’s also valuable to sketch what Soviet science looked like, at least from an outsider perspective. Bernal rightfully admired many Soviet scientists, but he turns a blind eye to the actual functions of the system. Bernal wrote in a 1941 Nature letter

In the creation of a new world it would be idle to look for the quietly pursued excellence and sound and acute scholarship that characterize an old established and stable society. Scier;ce in the Soviet Union is not like science in Denmark or SWitzerland. Nevertheless, the contribution to the advancement of knowledge made by the Soviet Union has been marked and important. In certain fields, especially in technology, Soviet discoveries and applications have already made a deep impression abroad. (360-61)

See Chapters 1 - 3 of Krementsov’s Stalinist Science for reflection on the 1920s and 1930s; Joravsky’s Lysenko Affair is a valuable examination of genetics in the USSR, but to your question is not entirely credible - it swings a little too far in the other direction.

Most curious is his preservation of faith after 1956:

Sage had a chance to rehearse his lectures at Oparin’s institute during a visit to the USSR in September 1956. It was an unsettling time for Stalinists – the waves from Khrushchev’s sudden indictment of the cult of personality and of the Stalin-era state crimes were rocking the previously unshakable Soviet edifice... Although Bernal was still treated as an honoured guest, he commented to Mackay that his lavish suite of rooms at the Hotel Moskva was poor compared to the accommodation on his previous visits, when his sitting room had not only contained a grand piano, but a stuffed bear proffering a box of matches (Brown 373-4)

In Marx and Science, Bernal writes

Science appears to the scientist, as well as to the ordinary member of the public, no longer as a hopeful and beneficent force, but as something which is willy-nilly being used for increasingly futile or destructive purposes. It becomes more and more difficult to think of science abstracted from society... (37)

And

Socialist planning of science is often distorted and caricatured as an attempt to plan thoughts and inventions in advance. No such attempt has ever been made, and if that had been the best use they could make of science, it would have been impossible for the Soviet Union to have achieved the results in peace and war that it has, in the face of enormous initial poverty and repeated foreign armed intervention. What really happens in the Soviet Union and the New Democracies is that science is applied to the solution of problems arising out of the general economic plan. For example, in the great combined schemes for the southeast of the Soviet Union, which are to change the whole face of nature and provide food for a hundred million people, some thousands of scientists of the most varied categories from mathematicians to archaeologists are studying, on the spot and in their laboratories... (43)

One of these quotes shows Bernal’s great insight; the other, his naivete. Turning to Science in History, on pretty much every page you can see Bernal discuss the relationship between capitalism and science. He did have some wonderful insights and predictions here, but he also crammed in 13th century technology and Grecian mathematics to fit his view.

In the introduction to the third volume, he turns more to monopoly and consolidation, along with the relationship between the state and monopolies, to discuss modern science. Again we have this eclectic mixture of quixotic faith in the Soviet Union despite, by this point, overwhelming evidence to the contrary, with enviable perception to the problems of postwar science.

Nevertheless...the importance of the developments in the socialist countries is far greater than their mere scale would indicate. They represent a new kind of way of employing natural and human resources which is impressing the workers of capitalist countries and even more the peoples of the under-developed countries. These have won some measure of political freedom and are now demanding effective economic liberation, an additional and powerful element in the transition from capitalism. (701).

That makes a short answer difficult - Bernal’s Marxist approach is not en vogue, and he does make generalizations to create a narrative that few historians would accept. He is naive about the realities of ‘Socialist science’ [b]. But he very presciently critiques postwar science, and is significant in both introducing the sociology of science, broadly, and challenging internalist histories of science. You should read Bernal, but not blindly, to develop your own opinions as to the value of his method and his place in the canon!