Latest annual statistics show that only 43.1% of trains to Tweedbank turn up on schedule.
South Scotland Labour list MSP Claudia Beamish is now demanding action to provide a more reliable service for passengers.
Her call comes as a petition signed by more than 14,000 people urges the Scottish Government to axe operator Abellio.
The ScotRail statistics show that services are late more often than not on a third of all rail routes across Scotland.
The Southern has reported frequently on problems with delays and cancellations affecting the new Borders Railway.
And the Scottish Labour party has published a dossier called ScotFail: How the SNP took our Trains off Track – and it is demanding action to improve rail reliability.
Ms Beamish said: “Most passengers accept that there are times when disruption can’t be avoided, but people shouldn’t have to put up with services that arrive late more often than they arrive on time.
“The fact that only 43.1% of trains into Tweedbank are arriving on time is unacceptable.
“The SNP transport minister needs to get a grip of what is happening on our railways and explain exactly what will be done to tackle delays and improve reliability.
“I have approached the minster about the cancellations and delays on the Borders Railway line, and I am waiting for his response.”
She had written to him about supporting the reopening of lines with a viable business case, such as Reston and East Linton on the Edinburgh-to-Berwick line to make the rail network accessible to more people.
She added: “The reality is that while passengers are continuing to pay more for a shocking service, Abellio has made £1m a month in profits from its franchise agreement with the SNP Government. The current set-up can’t possibly provide value for money for passengers in Scotland.”
Transport minister Humza Yousaf said he had called on the operator to draw up an improvement plan.
“ScotRail’s performance this year has not been at the level the Scottish Government demands and expects,” he said.
“Transport Scotland and Scottish ministers will be working closely with the franchisee to monitor delivery of this plan and ensure its targets are met.”
ScotRail did not dispute the dossier’s figure, but it said that was for trains arriving within 59 seconds of timetable time.
Using the industry standard measurement of within five minutes of the timetable time, its punctuality rate for Tweedbank was 86.1%, it said.
A spokeswoman said: “We have had a challenging few months due to industrial action, the big infrastructure works at Queen Street and the programme to electrify the line between Glasgow and Edinburgh. This has undoubtedly contributed to the fact that we are slightly behind our demanding target for performance.
“We have agreed an action plan with Transport Scotland about how we pull this back up to the level we would expect.”