What could it mean when there is red around the blue ink of a new tattoo but not around the black ink?

It could mean you are having an allergic reaction to the blue ink. It could also mean that your tattooists ink was contaminated. It's best to return to your tattooist and seek his/her advice. If you aren't comfortable doing that, seek the advice of a physician.

yeah i would say u are having a reaction to it, especially if its a nicer brighter blue