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How much deeper does the voice need to be to make a difference? Apparently 22.1 hertz which is approximately 1/6 the difference between the pitch of James Earl Jones (low-pitch) and Mike Tyson (high-pitch).
Other research (see below) have found that a deeper voice also makes a candidate more likely to win, are more desirable to women, mate more frequently, and father more children.
How can a man with a relatively higher-pitched voice achieve a natural-sounding deeper voice? The first and perhaps best way to accomplish this is through voice therapy. Think of it like singing where a person learns to achieve a greater vocal range through intensive instruction... but in this case, focusing on the lower range.
Of course, there are surgical interventions that may also be helpful should voice therapy be unsuccessful including:
1) Vocal cord bulking through injection
2) Cricothyroid muscle dennervation through botox or lysis
3) Thyroid cartilage framework surgery (Isshiki Thyroplasty Type 3)
I should point out that surgical intervention is NOT recommended as a first step and usually does not achieve the profound improvement that most men desire... in other words, there is no simple, quick, and easy way to deepen the voice.
Voice pitch and the labor market success of male chief executive officers. http://dx.doi.org/10.1016/j.evolhumbehav.2013.03.001, How to Cite or Link Using DOI
"Sounds like a winner: Voice pitch influences perception of leadership capacity," Proc Biol Sci. 2012 Jul 7;279(1738):2698-704. doi: 10.1098/rspb.2012.0311. Epub 2012 Mar 14.
Preferences for very low and very high voice pitch in humans. PLoS One. 2012;7(3):e32719. doi: 10.1371/journal.pone.0032719. Epub 2012 Mar 5.
Voters 'like low-pitched voices'. BBC 11/14/11
Your Voice Could Be Costing You Hundreds Of Thousands Of Dollars, Study Shows. Forbes 4/18/13