Let's be honest, a lot of things in life a carcinogenic to at least some degree, dependent on the dose and your unique biology. For example, I've inherited some bad genes from my father that result in an abnormally high amount of moles and new ones seem to appear most often in places of significant skin damage, so tattoos are out of the question for good, obviously.
But for other people the cancer\toxicological risks may involve less obvious genetic conditions like abnormal reactions to the ink itself that you've mentioned. Thankfully we already have initiatives that allow submitting your genome for analysis, so that's at least one factor you can manage.

Now ink and tattoo parlour hygiene are obvious concerns that so far we're only able to judge based on reputation and immediate inspection. It would be nice if there was at least a consumer watchdog that could provide some recommendations on available inks.

Even better, since body modification scene is only set to advance further, cooperation between licensed medical professionals and artists in the same space could be the best available solution. The biggest barriers to this are traditional cultural perceptions on both sides.