Final answer:
The United States began constructing its first destroyers in 1895, following the initiatives proposed by Alfred Thayer Mahan in 1890 and the subsequent passage of the Naval Act of 1890.
Step-by-step explanation:
The construction of the United States' first destroyers began in the late 19th century. Heeding the advice of Alfred Thayer Mahan in his influential 1890 work The Influence of Seapower upon History, the United States initiated efforts to build a stronger navy. The Naval Act of 1890 catalyzed the production of a new, modern fleet, and by 1898, the U.S. amassed a significant number of newly constructed steel naval vessels. Within this context, and considering the timing of naval armament expansion, the construction of the first U.S. destroyers began in 1895, which aligns with the option 1 in the question provided.