There are a lot of “turning points” in F1′s history. Any of them could be defined as the beginning of the modern era. Take your pick.
1950: The beginning of F1 itself. After all, there have been championships before F1, so F1 in and of itself can be the beginning of the “modern era”.
1958: First season without Fangio in it, first title won by a rear-engined car.
1968: Death of Clark, birth of sponsorship (Gold Leaf Lotus), the Cosworth DFV engine, and aerodynamics with wings.
1978: Aero grip moves ahead of mechanical grip in importance with the birth of ground effect.
1983: First season without Gilles in it, first season after ground effect ban.
1988: Beginning of Prost-Senna rivalry at McLaren.
1994: First season after Prost’s retirement, death of Senna, first title won by Schumacher, last title won by Ford.
2000: Schumacher’s first title with Ferrari.
2003: First major change in points system since the 60s, beginning of parc ferme rule after qualifying.
2007: First season after Schumacher’s first retirement, debuts of Hamilton and Vettel.