In the United States, is there an income or tax threshold above which an individual is required to make their estimated quarterly tax payments by electronic means?
No, it is encouraged, but not required.
From the 1040-ES instructions:
The IRS cannot accept a check for $100,000,000 or more. If you need to pay more than that amount by check, you must use more than one check. Please consider a different method of payment other than check if the amount of the payment is over $100,000,000.
No, there is no threshold above which electronic payment of estimated taxes is required of individuals; but corporations might be subject to different rules.
Electronic payment is encouraged since whether the estimated payment was made in timely fashion, or whether the post office lost the piece of mail or postmarked it the next day etc. are no longer an issue, but electronic payment is not required of individuals at any level of estimated tax payment.