Miyazaki for developing animation while most people are just obsessive with live action. Truly a pioneering mind. Even many Sci Fi and fantasy live action is heavily influenced by Miyazaki.
However, if you are talking live action it is Akira Kurosawa. Perhaps the single most influential over modern movies. Although there are important ones very early like Fritz Lang, whose influence is enormous as well. Welles is important for developing breakthroughs in Hollywood direction and tropes but overall his influence is greater over mid 20th century stuff than contemporary if compared to Kurosawa. Hitchcock is tremendously important in developing psychological thrillers and horror but his influence is not as wide as Lang or Kurosawa. Ozu is very influential as well, his directimg style heavily influemced not only film but comics as well.
Also, Kurosawa is the biggest influence on Miyazaki. Kurosawa is also the most often mentioned name when someone talks about a "great" director.
And the greatest artistic live action director is Tarkovsky.
So, overall the single most influential director might be Lang or Kurosawa. Though it's very hard to say, there are many other very i,portanto I didn't mention. I guess one should think too hard about BS like this and just enjoy movies.
Although in animation the greatest is Miyazaki, without any question. Thing is animation is mostly concentrated in a single country and Miyazaki is the king there. Live action is spread out through the world so it's hard to say. Usually people just list their current personal favorites.
I placed my bet on Kurosawa becuase he is without a doubt the most influential Japanese director. Since the most influential countries are France Japan and US, and only in Japan we have a unquestioned king I guess he tales the crown.