older red inks used to to do that on some people, but modern good quality reds these days hardly do it, I have not had a reaction in 35 years, and I have gotten lots of red since then with no reaction, and if you do get a red ink reaction that's pretty much all it is, maybe scab a little, not an infection.
I wouldn't worry about it, if the tattooer you have chosen has a clean shop and clean practice, go get the tattoo .