On another note, a war monitoring group said there had been no shelling by insurgent groups in that area for more than a week.
Hezbollah is fighting in Syria, backing the government of President Bashar al-Assad against a range of Sunni groups including Islamic State and the al Qaeda-affiliated Nusra Front.
Damascus airport and its surroundings are controlled by the Syrian government and allied forces. Between it and government-held central Damascus, rebels control a portion of the Eastern Ghouta suburb, which has experienced fighting for most of the conflict now in its sixth year.
"There has been no recorded shelling or firing from the Eastern Ghouta area onto Damascus International Airport for more than a week," Syrian Observatory for Human Rights director Rami Abdulrahman told Reuters.
Hezbollah's statement did not say when the attack took place or when Badreddine died.