This idea that Cuba was only important post-WW2 is a complete misconception, though an understandable one given that its importance isn't really explained to non-specialists. I'll keep my explanations limited to US-Cuba relations.

Cuba has been important to the US economy since before independence. Attempts by Britain to make its North American colonies buy more expensive molasses in the British West Indies was an important cause of aggravation in the lead up to the Revolution. John Adams wrote to a friend in 1818 that "I know not why we should blush to confess that Molasses was an essential Ingredient in American Independence. Many great Events have proceeded from much smaller causes." [there are more reputable places to find the quote, like Herminio Portell Vilá's 'Cuba in its Relations with the United States and Spain', but I'm trying to source statements with information that you can actually access]

In the 19th century Cuba became an increasingly important trade outlet not only to the US but also to Britain and France, even becoming the greatest producer of sugar in the world for a time (a role that it has intermittently occupied since, but never like in the 19th century). Sugar was an extremely lucrative business at the time, especially before effective use of sugar beats allowed for an alternative domestic production of sugar in European countries. Revenue from exports of Cuban sugar helped sustain Spain as at least a secondary power through the 19th century.

The reason that Cuba was so important to the US, however, was geographic. The US was moving West and in an age before trains this meant that communication with its inland territories as well as the ability to produce competitive exports from those regions required waterways. The Mississippi basin empties into the Gulf of Mexico, but the latter has only 2 exits: the Florida Straight and the Yucatan Straight, both of which are between Cuba and the mainland. If a major maritime power controlled Cuba (like Britain) were to take over the island they could effectively cut half the country off from the international market. All it would take is plugging up two narrow straights. Thus whoever dominated Cuba could hold much of the Western US at its mercy. [Daniel Walker Howe's great book What God Hath Wrought describes the poverty of US infrastructure pre-1848 and the importance of the Mississippi]

Even after the advent of trains, the Mississippi was still a major means of transportation for US exports. In addition, Cuba could be used by Britain as a staging base for a continental blockade, together with Canada. During the War of 1812 British ships used Cuban ports to get water and supplies as they hunted for American vessels. [Roland Ely mentions this in a footnote to his book Cuando Reinaba su Majestad el Azucar (not sure if this book was published in English or not, I've only seen Cuban and Argentine editions)]

In the 1820s Cuba was almost the object of war between the US, Great Britain, France, and Spain. George Canning pondered invading Cuba in 1827 to compensate for Absolutist Spain temporarily occupying Portugal. John Quincy Adams and others knew that they'd lose a war with Britain if they tried to take Cuba, but at the same time they were willing to fight to the last if Britain tried to annex the island or use its influence over Madrid (thanks to the massive debts of Spain) to try and buy or in some other way legally acquire the island. It almost sparked a war, so I'd say it was center stage then. [the best book on this is still Herminio Portell Vilá's Historia de Cuba en sus relaciones con los Estados Unidos y Espana but I'm sure many more general books on Cuban history at least touch on these issues]

From the collapse of the continental Spanish colonial empire, those powers seeking to maintain stability in the Caribbean (especially Britain) were very concerned with keeping Cuba Spanish. Even though it was no longer a major power it was better that Spain had the island than if France or, heavens forbid, the US of A were to acquire it. The US was obsessed with making sure other powers, like France or GB didn't acquire it for the reasons stated above. Spain was obsessed with keeping it for financial reasons. All in all, it was extremely important.

After the Spanish American War Cuba was extremely important once again, since though it was not a colony or possession of the US, it became an important base and staging point for the American commercial and military presence in Latin America. It gave Guantanamo over to the US as a military base for us to refuel our coal based military vessels that were the backbone of 'big stick' gunship diplomacy.

One of the major demands of the US before they left Cuba in 1902 was the inclusion of an amendment to the Cuban constitution (the Platt amendment) that basically said the US could intervene in Cuba whenever the Washington thought it would be in the best interest of Cubans (so, whenever we felt like it). We would use the amendment several times over the years until it was repealed by the 1933 revolution. The fact that we obliged them to include it at all is still a point of resentment among Cubans.

It was the US's most important foothold in the region and its economy had become (by the beginning of the 20th century) wholly dependent on the US. However by this time the US was buying cheap sugar from multiple resources and Cuba was less and less important to the overall US economy (though strategically it was still extremely important).

During both world wars Cuba was a major source of sugar for the US and its allies. In addition, as the US slowly displaced Great Britain in terms of commerce with Latin America and as the major military and diplomatic power in the region, Cuba was a key strategic ally. Cuban politicians could be trusted with advancing US interests in regional organizations, like the Organization of American States.

The US basically took for granted that Cuba would be at the very least neutral to American interests in the region. Then the Revolution happened.

After 1959 Cuba became what US politicians always feared: A staging point for a major foreign power (in this case the USSR). In addition, Cuba became a major obstacle in Latin America. The Cuban government trained and sometimes even helped arm guerrilla wars in Latin America whose leaders invariably wanted to strictly limit US government. Cuba also became a terrible example that countries in the region could stand up to the US instead of fearing that their policies would lead to invasions (Dominican Republic) or US backed coups (Guatemala). Cuba also successfully seized (with impunity) the property of US citizens. On principle alone the US felt this just could not stand.

The revolution had the added negative affect on the US of creating an extremely large Cuban exile community in Florida (which is a major swing vote in national elections). As such it gave well organized and impassioned exiles a level of influence other exile communities could only dream of. Thus foreign policy towards Cuba had to now pass through the additional lens of 'will normalization kill our chances to win Florida in the next election'.

So, in conclusion, Cuba has been regionally important to the US for centuries. You'll note that I don't even go into why Cuba was key to Spain's interests in the region prior to the 19th century (in which case it has been important for several hundred years more). The apex of Cuba's importance on the world stage has only come after 1959 (Cuban Missile Crisis, Cuba as a major regional and world political player, military interventions in Africa, aid to guerrilla fighters around the world, general thorn in the side of US foreign policy, etc.). If phrased in terms of US history our relations with the island have been important since the beginning.

Great overview of US-Cuba relations: Lars Schoultz: That Infernal Little Cuban Republic.

Note that due to our 20-year rule any answer would have to omit the recent past and current events.

Still there's plenty be said about Cuban-American relations in the post-WW2 era up to 1995.