Question

Based on this map, what made Cuba such a serious threat to the United States during the Cold War?

Answer 1

The thing that made Cuba such a serious threat to the United States during the Cold War was its close physical proximity to Florida, since this meant that missiles could be launched from Cuba and hit the US. 

Answer 2

The map of Cuba and the US in the Cold War era is attached in this answer.

Cuba was a communist country and an important ally of the Eastern bloc leaded by the URSS during the Cold War. The main threat posed by that alliance to the US (leader of the Western bloc), was the fact that Cuba is really close to the US territory, specially to Florida, and that misiles shot from Cuba could reach the US.

The best example were the tensions that took place in 1962, when the Soviets deployed ballistic in Cuba, that were close to leading to a full armed conflict. This was actually called the Cuban Missiles Crisis.