Henry Drexler was born on August 7, 1901, in Braddock, Pennsylvania, and grew up in Bethany Beach, Delaware. He attended Culver Military Academy in Culver, Indiana, and then entered the U.S. Naval Academy on May 29, 1920. Midshipman Drexler performed his summer cruises aboard the battleship USS South Carolina (BB-26) from June to August 1921, aboard the battleship USS Delaware (BB-28) from June to August 1922, and aboard the battleship USS Arkansas (BB-33) from June to August 1923. He graduated from the U.S. Naval Academy and was commissioned an Ensign in the U.S. Navy on June 4, 1924, and was under instruction at the Third Naval District at the Navy Yard in New York City from May to October 1924. Ensign Drexler reported aboard the light cruiser USS Trenton (CL-11) at the Washington Navy Yard in Washington, D.C., on October 4, 1924, and he was killed in a powder charge explosion aboard the ship on October 20, 1924, for which he was posthumously awarded both the Medal of Honor and the Navy Cross for his heroism in attempting to save his crew members. His Navy Cross was awarded by the Secretary of the Navy on November 18, 1924, and his Medal of Honor was awarded by a Special Act of Congress on February 3, 1933. Ensign Drexler was buried at Arlington National Cemetery.

His Medal of Honor Citation reads:

For extraordinary heroism in the line of his profession on the occasion of a fire on board the USS TRENTON. At 3:35 on the afternoon of 20 October 1924, while the TRENTON was preparing to fire trial installation shots from the two six inch guns in the forward twin mount of that vessel, two charges of powder ignited. Twenty men were trapped in the twin mount. Four died almost immediately and ten later from burns and inhalation of flame and gases. The six others were severely injured. Ensign Drexler, without thought of his own safety, on seeing that the charge of powder for the left gun was ignited, jumped for the right charge and endeavored to put it in the immersion tank. The left charge burst into flame and ignited the right charge before Ensign Drexler could accomplish his purpose. He met his death while making a supreme effort to save his shipmates.