The rail operator advertised 18 trainee posts but was swamped with 2229 job requests by January's deadline.
It means applicants have a one-in-123 chance of landing one of the coveted positions on the 35-mile track which will link Edinburgh Waverley with the Borders region.
Christine Grahame, MSP for Midlothian South, Tweeddale and Lauderdale, and a long-term supporter of the scheme due to open in 2015, said the high interest in the Tweedbank-based jobs was a good sign.
She said: "They are very attractive jobs.
"People want to be involved in a new project with a national profile such as this.
"The positions carry a bit of kudos. It will be great to say you drive a train for a living.
"I also think it is good for the area, as successful applicants will likely be from around the country and therefore will come to the Borders to live."
The posts have a starting salary of £22,281, rising to £39,204 after training and when the probationary period is completed.
A ScotRail spokeswoman said: "The selection process has begun and the first recruits will begin their training in late spring."
Network Rail also announced this week it has employed 15 local people to work on the line, with three junior posts being created with main contractors BAM Nuttall.
Planning applications for seven new bridges at Glenfield Road, Wheatlands Road and Winston Road in Galashiels, as well as Falahill, Heriot, Hanging- shaw and Fountainhall, have also been submitted.
Construction work on the Borders Railway – which will replace the Waverley line that was axed in 1969 as part of the Beeching cuts – is due to start this spring and is to be completed in 2015.
It will have 10 stops – Edinburgh Waverley, Brunstane, Newcraighall, Shawfair, Esk-bank, Newtongrange, Gore-bridge, Stow, Galashiels and Tweedbank – with the total journey time estimated at 55 minutes.
Meanwhile, ScottishPower has received more than 1000 applications for 80 apprenticeships following a recent recruitment campaign.
The company has been looking for apprentices to fill places across three programmes to help deliver its £3.6 billion investment drive in the UK.
Two of the three programmes are now fully subscribed, but it is still looking for up to 42 young people aged 16 to 19 for its Engineering Foundation Programme.
The scheme combines study at a local college with industry-specific training at a ScottishPower centre.
Positions are available in Edinburgh, Glasgow and Chester.