I'm sorry if this is a bit of a stupid question, but my understanding of a Banana Republic in the classic sense was that they were countries extremely dependent on a small number of exports, usually agricultural products, with these markets and industries being heavily controlled by oligarchic, extremely powerful international business enterprises that often originated in the United States and had vast amounts of economic and political clout to force these countries to play by their rules, with the backing of the American state to do so, in this case most notoriously the United Fruit Company. All of this would force these countries into unfavorable trading relationships and stymie their political and economic development.
I wouldn't have thought that Bananas specifically would be such an important crop that they would lend their names to exploitative political-economic systems, certainly they seem like a luxury product in most countries that are nice but not really a big deal if they aren't available. How big a market was there for bananas specifically, why was it so in demand and was it the one crop that dominated the economies of entire countries across Central America and the Caribbean in the way that the term implies?
It wasn't so much that bananas dominated the economies of these countries, but more the great influence of the United Fruit Company (UFC) in their economies and politics. Only in one of the banana republics (Honduras) were bananas the top export (in about 1930, bananas were over 50% of exports from Honduras by value). In the other countries in Central America at the time, coffee was the dominant export (in El Salvador, Guatemala, Nicaragua, coffee was over 50% of exports by value, and in Costa Rica, coffee exports were much greater in value than bananas). However, coffee plantations were usually locally owned (by rich landowners), while banana plantations were mostly US owned (and mostly by the United Fruit Company, which dominated the US banana import trade, peaking at over 80%). Foreign companies still took the biggest share of the profits from coffee, due to controlling processing and transport. UFC was a major owner of transport infrastructure in Central America, building (and owning) railroads and port facilities and ships. Apart from earning coffee profits from transport, UFC's railroad ownership was used an an anti-competitive tool, with UFC's banana cars charged much lower rates than those of competitors (which eventually cost UFC a 1959 court judgment in New York, and they had to pay US4.5 million in damages to shareholders of the railroad). UFC's lesser US competitors in Central America were also in the railroad business.
UFC's influence went beyond transport, with UFC establishing the Tropical Radio and Telegraph Company, and running the Guatemalan postal service. UFC helped negotiate US loans to Central American governments.
So, why "banana republic"? There were three main issues that entangled UFC and its competitors in Central American politics. First, their railroad concessions and land ownership depended on government favour. Second, labour relations could be managed better with government favour (and the police/army on UFC's side in any labour disputes). Third, the size of UFC made it an important player in politics - for example, in Guatemala, where coffee dominated exports, UFC owned almost all the railroads, and was the biggest landowner and the biggest employer. In 1930, UFC's annual profit (approx US$40 million) was much greater than, e.g., the budget of Honduras (approx 15 million pesos = approx US$8 million). UFC and the other big banana companies needed government favour, and were big enough to obtain it. Bribery and threats were used. Not for nothing was UFC nicknamed "El Pulpo" (the octopus) in Honduras - it's tentacles were visible almost everywhere. They were also unafraid of trying to influence the US government, lobbying successfully for the US backing (via the CIA) of the 1954 overthrow of the elected Arbenz government in Guatemala (UFC saw Arbenz's planned land reforms and labour reforms as serious threats). Thus, even where coffee ruled the economy, UFC pervasively influenced the nation and politics.
EDIT: Honduras, with its banana-dominated exports, inspired the term "banana republic", which first appears in O. Henry's 1904 book Cabbages and Kings, to describe the Honduras-inspired fictional country of Anchuria. (O. Henry (William Porter) lived in Honduras for 6 months in 1897, and wrote Cabbages and Kings while in Honduras.) From this brief literary usage, the "banana republic" acquired its more political meaning, that of countries wherein politics is dominated by an unequal relationship with powerful foreign companies. The term "coffee republic" is sometimes used for the Central American countries with coffee-dominated exports, and "sugar republic" for Caribbean countries with sugar-dominated exports. Both terms have historically referred to the same kind of political dominance of foreign companies - the economic might of UFC even in the coffee republics is briefly described above, and in the sugar republics, the sugar plantations were largely owned by large foreign companies which exerted similar influence to UFC.
Further reading:
The United Fruit Historical Society has a nice chronology on their website, detailing many of their ups and downs in Central America: http://www.unitedfruit.org/chron.htm