Who had the most influence on modern music? That's easy. The Beatles. They took a new, emerging sound from artists such as Chuck Berry, Carl Perkins, Elvis Presley, Jerry Lee Lewis, etc. and made it their own. They used these influences to mold their song writing craft and produce music that has arguably influenced all mainstream pop / rock genres. Not only this, but they also brought this music to the largest possible audience. This new music was reaching more people and affecting them more than any other artist of the time.
Their mark on music can be heard in the works of other artists, from Eric Clapton, ELO, and Led Zepplin to Michael Jackson, Oasis, The Eagles, etc.
An interesting research project might be to read through liner notes, interviews, etc. and actually figure out which person or group has been cited more as an influence. I don't want to be the one to do this though.