I'm not a millenial. Nor are any of the attendings, fellows, etc. with whom I've worked who sport tattoos.
The world is changing though. Once upon a time, only blue collar folks, criminals, and those who had served their country (also, usually from blue collar families) had tattoos.
Now, nice girls from upper class families have them. Middle class family guys have them. Grandmothers have them. Lots of people have decided that a little ink isn't a terrible thing. Patients certainly have them, and every interaction I've had with a patient who noticed one of my tattoos and wanted to mention it was a positive one. There may be some who are quietly, sullenly judging me, but I'll take that in trade for the hundreds I've connected with over a common personal interest.
You can try to cling to the past and insist that everyone live up to the standards of your childhood, and if you have the power to enforce that, you might even hold the line for a while. You, too, can pass over talented recruits because of something so literally skin deep, and they will find places elsewhere. Meanwhile, you will maintain your enclave of the past until you, too, are eventually replaced by someone with ink or who doesn't care if others have it or not.
They, meanwhile, will hold the line against whatever the next social trend may be. Same as it ever was. Those damned kids find a way to get on your lawn, generation after generation.