This will depend on the solver you’re using: in principle, anything accessible via their C API is available in julia/JuMP with a little code. To get the basis indices (active constraints and bounds), there’s a getbasis function in MathProgBase that you can use. Right now Clp, Gurobi, Mosek, and CPLEX have this wrapped. You can call it with something like MathProgBase.getbasis(getInternalModel(model)).
-Joey
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MathProgBase.getconstrmatrix(getInternalModel(model))