If $H$ is an infinite index subgroup of the braid group $\mathcal{B}_n$, is there a way to find a presentation for $H$ ?

Unless you have a specific type of subgroup, like one that acts cocompactly on something, you're in deep trouble. Braid groups are incoherent (see Artin groups, 3manifolds and coherence by Cameron Gordon). That is, there are finitely generated subgroups that are not finitely presented. 


I'll address the question in your comments, can one determine a presentation for a subgroup of the mapping class group generated by Dehn twists? Given a bunch of simple closed curves on a surface, there is a minimal essential subsurface containing the curves. Then the Dehn twists will generate a subgroup of the mapping class group of this subsurface, which will have infinite index in the mapping class group of the full surface if it is a proper subsurface. If it is finiteindex in this mapping class group of the subsurface, then there is a procedure which will terminate with a presentation. The problem is that the procedure may never stop if the subgroup is infinite index in this subgroup. The idea is to enumerate finiteindex subgroups and their generators via ReidemeisterSchreier, and then see if your elments can generate the generators for these subgroups. If you can, then you know your group is finiteindex, and you can test all intermediate subgroups to see if your elements lie in them by looking at their image in a finite group quotient. If the subsurface is planar, then the subgroup generated by Dehn twists about the curves will be a subgroup of a central extension of the braid group. If your curves are labelled $c_1, c_2, \ldots, c_k$, such that $c_i\cap c_j=\emptyset$ if $ij>1$, and $c_i\cap c_{i+1}=2$, and the subsurface is planar, then the group will generate the mapping class of the planar subsurface, since these are the form of the standard generators for the braid group (the central extension comes from Dehn twists around boundary components). For subgroups generated by 3 Dehn twists, see this paper. 

