One danger would be of going bankrupt. Nebulae are not super rich sources of matter. From wikipedia: "Although denser than the space surrounding them, most nebulae are far less dense than any vacuum created on Earth – a nebular cloud the size of the Earth would have a total mass of only a few kilograms." https://en.wikipedia.org/wiki/Nebula
Also nebulae are huge - "Most nebulae are of vast size, even millions of light years in diameter."
A second danger would be from radiation. While a Kardashev II civ is likely to have technology well beyond that needed to shield against mere isotopes, the debris from a supernova is highly radioactive.
The earlier you start mining after a supernova, the better. The matter will be less diffuse and it will be composed of more exotic isotopes.
I can think of a few ways to mine a nebula, but it's hard to imagine the economics of a K II civ. They have enough energy available to move their star! https://en.wikipedia.org/wiki/Stellar_engine They could send lots of mining ships into the nebula at high speed. The ships would be variants on the Busard ramjet design and would use magnetic fields to gather together the diffuse material. A much better idea would probably be to guide matter by throwing gas giant planets (or even stars) into the nebula. 99
The gravity well of these objects would tend to concentrate the diffuse gas. A third option would be to push some of the gas around using either the light or solar wind of their home star and focusing that into a beam which they can scan across the nebula as needed.
Honestly though, I can't think of any reason to bother. What could a K II civ want to make that would require more matter than is available in a few gas giant planets? The only thing i can think of would be a star, and stars are a dime a dozen. In order to get to the nearest nebula, you would probably have to pass by dozens of stars.
The only thing I can think of is that they might scan through a nebula looking for the blasted fragments of the planets that formerly orbited the star that exploded. There might very well be unusual objects left over from the supernova. For example, large carbon rich planets are hypothesized to have enormous diamond crystals inside their cores. It might be amusing to a K II civ to grab a moon-sized diamond crystal and use it for an embassy or something. "This old thing? Oh, it's just a moon-sized diamond that we carved into an embassy and towed into your solar system. It's no big deal." 99