How do we continue in the simplex algorithm if some artificial variable is basic ? Do we still forget about this basic artificial variable in the second phase of the simplex algorithm as we do forget about it if it is not basic?
If the current basic solution to your Phase-I problem is optimal and one of the auxiliary (artificial) variables is basic and strictly greater than 0, then this means that your original LP is infeasible. You're done.
If the current basic solution to your Phase-I problem is optimal with an objective value of 0 and an artificial variable is basic at zero, then you will need to drive this variable out of the basis with a degenerate pivot. Among the original (not artificial) variables that are non-basic in the optimal solution to the Phase-I problem, look for a variable that can enter the basis in place of the artificial variable. Repeat this process until you have a basis consisting only of original (not artificial) variables. Procedures for doing this are discussed in many textbooks on the simplex method. For example, you'll find this in Chvatal's Linear Programming.