One reason we see Max-CSPs more is that they often capture convex optimization problems for which minimization is easy (e.g., Max-CUT vs Min-CUT).
A Min-CSP is equivalent to a Max-CSP when you negate all the constraints. So minimization and maximization are equivalent if you're asking for the optimum solution. But this reduction is not approximation preserving in the sense that, say, a constant factor approximation of the negated CSP does not necessarily imply a constant factor approximation of the original CSP. Approximation of Max-CSPs and Min-CSPs can however be studied under the unified framework of "generalized CSPs", when constraints have weights and can be fractionally satisfied. Prasad Raghavendra's thesis is a good resource about this.
The specific problem of Vertex-Cover is not really a CSP in the standard sense. However the approximation of the problem is captured by the so-called "free-bit complexity" of PCPs. See "Free bits, PCPs and non-approximability -- towards tight results" by Bellare, Goldreich, and Sudan.