When you say "algebraic monodromy group" do you mean the etale fundamental group? In that case, you're really asking about the image of monodromy in the symplectic group mod p. At this point there are lots of techniques for showing monodromy is big; for instance, see Chris Hall's paper "Big symplectic or orthogonal monodromy modulo l"
http://arxiv.org/abs/math/0608718
where he proves, among other things, that a family of the form y^2 = f(x)(x-t) has big monodromy in this sense. That paper, the paper it cites, and the papers that cite it should give you good ideas about how to approach the computation of mod p monodromy for any particular family of hyperelliptic curves you might encounter.
On the other hand, you might have in mind the topological question about the monodromy of a family of holomorphic curves in the discrete group Sp(2g,Z), which is somewhat more subtle. It can be the case that the monodromy surjects onto Sp(2g,Z/p) for every p but has infinite index in Sp(2g,Z); this is called the thin case. Actually, for y^2 = f(x)(x-t) one can prove that the monodromy is big even in this stronger sense; this was first proved in unpublished work of JK Yu, and I give another argument in Example 5 of my expository paper "Superstrong Approximation in Monodromy Groups":
http://arxiv.org/abs/1210.3757
which also has some general discussion of thin and non-thin monodromy groups.
I learned the argument in that paper from Ian Agol as a result of asking an MO question about it:
The image of the point-pushing group in the hyperelliptic representation of the braid groupThe image of the point-pushing group in the hyperelliptic representation of the braid group
