It's been a common refrain for decades, and in fact a lot of the pundits of the period liked to talk about it, but the actual evidence is pretty thin.

The best way to determine who "won" a presidential debate is probably to conduct a poll and ask people who they think won. So what we're looking for is a poll that specifically breaks out people who watched the debate vs those who listened to it, while controlling for other factors.

Unfortunately, that poll doesn't really exist! The closest we can come is a subsection of a larger poll conducted by Sindlinger and Company. It shows Nixon with a slight edge among radio listeners and Kennedy with an edge among Television viewers, and noted that the Television audience is larger to boot. But it's flawed in other ways.

Anyway, here's an article about it: http://www.pollster.com/blogs/did_nixon_win_with_radio_liste.html

which also notes that a University of Minnesota political science professor did a study attempting to replicate the findings, and in fact did so. There's further sources beyond that, but some of them are behind paywalls, because academia is annoying.

How would you define "winning" a presidential debate?