I don't think traditional mining would be possible at all.
Consider that all known elements are created by stars - the bigger the star, the bigger the element they can create. These elements are not created in a solid state, this happens after the element has cooled, but in the core of a star the elements are what can best be described as a gas or liquid.
And this would be the same for any element you tried to use on the surface of or inside any star - the metals and alloys would melt and become unusable as mining implements.
So mining a star would be impossible.
What would you mine a star for?
Well considering the technology level of your Borlaks, they would probably find rarer elements to be more expensive and valuable. Hydrogen and Helium are plentiful in space. Any space faring race capable of dabbling with anti-matter and scouring entire planets for resources would be able to collect these elements in vast quantities with ease.
But Hydrogen and Helium are the smallest elements. The larger the element, the rarer it becomes - Uranium for instance, one of the largest elements, can only be created in the very largest of Red Giants, making it one of the rarer elements in the universe - so there may be some financial gain in harvesting larger elements from Red Giant Stars, instead of having to wait the (potentially) millions of years before these gigantic stars go supernova.
So now that we have established not only that mining wouldn't work, but there would be good reason to "mine" elements from stars, what are the options of the Borlak race?
I would suggest siphoning the star. Literally sucking part of the star away into space, so the elements from the star are colder and can be harvested in a cooler environment. By sucking up the contents of a star the Borlaks could get at the bits they needed without having to operate within the extreme temperatures of a star.
The only problem is the immense gravity of a Red Giant. The gravitational pull of a star that big would be far greater than any "space-vacuum" the Borlak's could develop.
However, there are things out there with a greater gravitational pull than Red Giants...
If the Red Giant star is close to a Black Hole, the Black Hole will do the siphoning for you. All the Borlaks have to do is ensure that they don't fall into the event horizon and they should be able to collect parts of the Red Giant Star as it is ripped away and plunges into the Black Hole.
Of course, this wouldn't be easy. The Red Giant star would have to be very close to a Black Hole for this to happen, making life very difficult for the Borlaks - falling into the event horizon would be a huge risk, and once they do there would be no escape.
Also, this wouldn't be plausible for just any star - only Red Giants would make the elements you wanted.
And this wouldn't be plausible for any Red Giant star either - you would need to find one that is right next to a Black Hole.
But with the right conditions, and with enough care and sufficient technology to pull it off, it may be possible to harvest what you want from a star in this fashion.
Of course, if the Borlaks were even more advanced than this, they could be carrying around their own portable Black hole with them to help them siphon off Red Giants - however if they were advanced enough to do this, they would probably be advanced enough to create the elements they wanted manually, without having to mine them at all.