This is really a tricky question...
Is it the most recognizable name (with today's folks) that's been associated with Broadway (ever)?...
...Or the most recognizable person (with today's folks) that is KNOWN PRIMARILY as a Broadway name?
I think you'll find different answers for both:
If it's Choice #1 - I would say Bob Hope, Marlon Brando, Glenn Close, Katharine Hepburn, Bette Davis, Denzel Washington, Nicole Kidman, Angela Lansbury, Paul Newman, Madonna, and maybe a few legendary composers like Gershwin, Berlin & Porter, etc.
If it's Choice #2 - I would say Ethel Merman, Rodgers & Hammerstein, Stephen Sondheim, Bernadette Peters, even Neil Simon, and a few others...
And jasobres --- There's an old saying about stardom that's pretty true (although budgets have changed a lot on TV since this first was batted about)...
Movies make you rich...
TV makes you famous... (more people watched three episodes of the Cosby Show when it was "hot prime time" than all the people worldwide who saw E.T. in its initial release)
Theatre gives you clout.
"Jaws is the Citizen Kane of movies."
blocked: logan2, Diamonds3, Hamilton22