Sunday services - which run only once an hour - will be doubled to four carriages from this weekend.
Peak-time trains on Saturdays, when trains operate half-hourly, will be similarly lengthened.
Extra carriages will also be added “in the short term” to the weekday 4:24pm Edinburgh to Tweedbank service, which ScotRail described as “particularly busy”.
The moves come after the new line’s first week was plagued by a catalogue of problems, including overcrowding and delays.
Passengers complained of being unable to board trains because they were so full, or having to stand for the whole one-hour journey on the 35-mile line.
Delays have increased because of the extra time taken for passengers to get and off crowded trains.
This has been made worse by the doors of the trains used being at the ends of carriages rather than in the middle, unlike those on some other busy routes.
ScotRail said the level of interest in the Borders Railway had been “remarkable” and thousands of people had turned up to travel, particularly at weekends with families making a day of it.
However, David Spaven, a rail consultant and author of Waverley Route: the battle for the Borders Railway, said the operator should have foreseen the high levels of passenger demand.
He said: “The level of interest in travelling on the railway wasn’t at all ‘remarkable’ – it was highly predictable.
“ScotRail were warned by campaigners, but chose to ignore well-informed advice.
“[Scottish Government agency] Transport Scotland and ScotRail have had years to plan and provide for the rolling stock needed to cater for demand on the new railway.
“Given that Borders trains serve not just seven new stations in Midlothian and the Borders, but also the well-established and popular Edinburgh stations at Newcraighall and Brunstane, the minimum requirement is surely four-coach trains on all peak-hour services on Mondays to Fridays, and ideally they should all be six-coach.
“Adding extra coaches to one daily train on Mondays to Fridays - out of 33 in each direction - just isn’t good enough for commuter needs.
“Longer trains on Saturdays and Sundays are of course very welcome, but it shouldn’t have needed the at-times chaotic performance over the last week for ScotRail to see that this was required.
“It was critical to get things right in the first few weeks of the railway, but the opportunity to build on all the goodwill towards the new railway has been partly squandered.”
Campaign for Borders Rail chairman Simon Walton said: “We’ve said all along there was much more demand than was being planned for.
“We are delighted the Borders Railway is already proving sceptics wrong, and that is to be celebrated.
“However, we raised the issues of underestimating demand with the Scottish Government and its Transport Scotland agency even before track laying commenced.
“The campaign was assured all trains would be lengthened in the initial weeks at least, but with many services running in the very minimum formation of two carriages, that is clearly not the case.
“Were it not for the general shortage of trains, adding more rolling stock would be relatively easy - although we would not countenance the inconveniencing of passengers elsewhere.”
A ScotRail spokeswoman said: “We are continuing to monitor passenger numbers so we can tailor how to make best use of the carriages available to us.
“For example, we may have found a way to temporarily double the number of carriages on a Sunday, and aim to do the same on Saturdays during peak times.
“We are also planning to add carriages in the short term to one particularly busy service, the 16:24 from Edinburgh-Tweedbank which is key for commuters.
“The Borders Railway is undoubtedly going to be very popular with commuters and leisure travellers alike, and we remain committed to achieving the best balance so that customers have an enjoyable experience.”
A Transport Scotland spokesperson said: “Seating capacity on the Borders Railway is at the level planned prior to the launch and there has been no reduction in the number of carriages operating on the route, however we do recognise the higher than expected demand has led to some pressure on the services and we are working together with ScotRail to address these issues.”