e2x is always positive. Therefore, its range is (0,+∞). However, as you figured out, arccsc(x) does not accept (0,1) as input.
What you need to find is the range of the values x such that e2x falls in [1,+∞) because if not, then arccsc(e2x) is not defined.
Can you solve the inequality e2x ≥ 1?