Gas giants are cheaper and require less handwaving
- Stars are made of plasma. If you harvest the material from the outer layers of a star - or even from ejected matter - you will need a way to cool it down. Space is devoid of the gases needed to move this heat elsewhere, so it will require enormous space radiators. This is very, very inefficient. Meanwhile, gas giants are composed of gas and small particles! No large-scale cooling necessary.
- Stars have more gravity. It takes much more work to orbit a star safely than it does to orbit a gas giant. For large-scale loads of matter, you will want to save energy.
- Stars are hot. Unless you use an unfathomably large magnetic field to move the plasma toward your ships (which would require so much energy that the magnets would melt), you will want to come in contact with the surface of the sun in some way. For reference, the surface of the sun is 5,777 K, while iron vaporizes at 3135 K: without handwavium, it's extremely difficult to collect any material.
- Stars have the wrong compositions. While gas giants may mostly be made of hydrogen and helium, they are theorized to have solid cores of more useful elements. In addition, stars have significantly more internal pressure - that will crush even the strongest apparatus. They literally function by ripping apart and smashing together subatomic particles - there is no plausible way to put anything deep into a star to harvest useful elements.
Even if you opt to fuse hydrogen to make heavier elements (as opposed to mining them) it is easier to acquire that hydrogen from gas giants. It's easiest to opt for the cheaper option.
##To mine a sun
To mine a sun
You need to have the energy to escape a star's gravity at virtually any distance; to construct materials impervious to heat that will normally turn any element into a plasma and tear it to shreds; to withstand some of the most extreme pressures in existance; and to cool a plasma to a gas en masse. Sounds like a Kardashev 2.9
##To mine a gas giant
To mine a gas giant
You need to have the energy to escape the mass of a gas giant; to siphon gas in large quantities and transport it, and to withstand a fraction of the pressure that a star would contain. Sounds relatively close to a Kardashev 2.