I don't know the full process for making these, but I can take a fairly educated guess.

First they are going to take the DNA sample and use a reaction called PCR to amplify it. This will copy a small section of DNA, not the whole sequence. For the PCR to work for all of their customers, this has to be a region* that nearly everyone has in their DNA.

They then take the PCR product and treat it with different enzymes that are like molecular scissors which will chop at specific sub-sequences. The personal nature of the art comes from small differences between people in the region amplified in the original PCR. Different sequence = different cuts = different lengths of pieces.

They then run the enzyme digests on a gel, which is like a slab of thick jello. The bigger the piece, the slower it moves so the pieces separate. The lanes on the sides are a standard "ladder" of known sized pieces. You can visualize these gels under a UV lamp.

They can either use molecular tags or, more likely, photoshop to make the art piece look more interesting.

There is not any actual sequencing of the DNA happening in this process, and the band patterns are pretty low-resolution so it's unlikely that this could be used to identify someone.

If this is a legitimate operation, then there is not a situation where the art company has an unsecured disk with lots of DNA sequences on it.

All that said, the concern about sharing genetic material with random companies is valid because they could also sequence it if they wanted to, but that would be relatively expensive and actively malicious. I believe the risk is low but non-zero, and everyone will have a different comfort level with that.

If you want alternative options you might be able to find an open lab or local college that will work with you to run your own PCR and gel, then photoshop the result yourself.

Hope that helps you make a more informed decision.

*To be a little more pedantic, everyone has to have the same ends of the region where the copying starts and stops. The part in between, which is a small fraction of your total DNA, can still be different.