If while getting a tattoo a couple of months ago the tattoo artist sexually assaulted me, do I have a case to pursue this in a civil manner?

You may be able to sue the tattoo artist himself, but it may not be worthwhile.

There are two elements to a successful lawsuit: liability and damages.

Liability is a wrongful act, either negligent (careless) or intentionally wrongful. Sexual assault would be intentionally wrongful, so from what you write, there would likely be liability.

But the civil law system (lawsuits) generally only awards compensation commensurate with the costs you incurred and/or lasting injuries you suffered--and while you can collect for emotional or mental damages in a case like this, it is always much more difficult to collect for them than for physical injury.

You do not describe what happened. But if you suffered little or no actual physical assault, with little or no physical injury and are not in expensive counseling as a result of the attack, then even though you are justifiably outraged and upset by what happened, a lawsuit might not award you enough money to justify the cost, time, stress, etc. of the suit. Lawsuits are about compensation, not justice, so there must be some injuries or costs to be compensated for. (Obviously, if you have or are incurring significant medical costs, suffered some long lasting injury, etc. that would be a very different story.)

You most likely cannot successfully sue the business: a  business is generally not liable for the intentional wrongful acts--especially criminal acts--of its employees since those are not committed "in the course" of employment: the artist, for example, was not hired to assault customers, and the business will not usually be responsible for crimes that are not part of  the job.