Gershwin is probably my favorite American songwriter/composer on any given day but the question is influential musician not greatest songwriter. Is Gershwin even the most influential songwriter of the jazz era? I don't think he is. That would probably go to WC Handy or maybe Scott Joplin. Their music is at the root of blues and jazz respectively.
Some of the answers are getting off point IMO and yes, this is purely a subjective exercise, none of it can be proven. A great case can be made for Elvis. How many ten's of thousands of impersonators has his influence begat? All seriousness aside he is a major influence as a musician? Yes, if you count a singer in the equation but then the question should be most influential singer? While a singer is technically a musician it never entered my mind to consider individuals know as singers (i.e. Sinatra, Presley, Fitzgerald, Vaughn, Holiday, Madonna etc) or songwriters (i.e. Gershwin, Porter, Arlen, Berlin, Dylan). I certainly consider many of the above excellent musicians but their influence is greater as
Singers and/or Songwriters. I was thinking along the lines of a musical instrument musician. Is that the intent of the question or do singers and songwriters count as well?
Leonard Berstein and Arthur Fiedler are probably the most influential personalities that brought orchestral and classical music to the masses. Does anyone remember Berstein's show in the 50's something like "Young people's guide to the Orchestra" I was greatly influenced by that show as a kid as I'm sure many more were. But for the question posed I don't think these guys count. The question requires narrower boundaries. What do you say Jeffery, its your thread?