I don't know any reference where such a presentation is written down for any Artin group. But:

 - You can find here a presentation for the pure Artin group of type B: http://arxiv.org/abs/math/0408035 Proposition 1.1 (by setting $N=2$ is the formulaes)
 - For all the infinite families, the corresponding pure braid groups are iterated semi-direct products of free groups. (For all the families but the $D_n$ one, it follows from the fact that the corresponding hyperplane arrangements are of fiber type, hence these are even almost-direct product). It should leads quite easily to a nice presentation of these groups.