Children in the Classroom
In my years of teaching in Asia I have had one class session where a student brought a child with him. It was an exceptional case but I was surprised he did not ask for permission. The child was well behaved (maybe 7 years old) and sat in the back not disturbing the class in any way. For this reason, I let it slide and I might be willing to accept it happening in the future.
However, I do make it quite clear to my students, I am the captain of this airplane and I will not tolerate ANYTHING which negatively impacts the learning environment. This includes anyone who disturbs the learning process in any way. I agree with Pete L. Clark - it is not a childcare issue. You need to focus the students on it being a learning environment issue. If a student does not turn off their ringing phone, out they go. If someone dresses in a way which distracts students or me, out they go. If anything exists which negatively impacts the learning process for even one of my students, out they go.
I'm pretty strict on this and I don't generally have problem because of that.
Back to your core question: How do you maintain a level of education while not being a jerk? You focus on the real issue. The real issue is not kids, the issue is disruptions. While you can be forgiving and understanding, to do so in a way which negatively impacts your students should never be accepted.
Most universities I know have both a cultural understanding and formal regulations that the only people who are allowed in the classroom are those that are registered for the course, except where explicitly permitted by the instructor. (Thus for instance one has the notion of "auditing" a course: this basically means that you are not signed up to take the course for a grade and will not complete the required coursework / take any exams, but you do have the instructor's permission to sit through the class meetings.) This is a defensible regulation: without it, who knows who would show up for a course, taking up possibly limited space and occupying the attention of the instructor and/or the other students?
Children are people, right? I would thus frame the discussion in that way: you're not discriminating against someone because they're a parent. You're just not allowing people in the classroom who are not registered for the course.
I am somewhat surprised that this is a problem for you at all, and I wonder where you are teaching and if the cultural mores and regulations are different there. I don't know of any American university in which people would think they could bring children to class except in some truly exceptional/emergency situation in which they have received the instructor's permission. In any case, I would advise you to look up your university's specific policy on "unregistered attendees". Assuming it is along the lines of what I am suggesting you should, at most, modify your syllabus to quote from and/or link to this general policy. Don't make the issue about child care at all.
Added: I just looked at your profile and saw that you say you are in South Korea. As I said, both cultural mores and regulations may well be different there, and if it is very common for students to bring children to class, that makes me much less confident that rules or customs are being violated. So to adjust my answer for this: "Do other universities have policies about children in the classroom?" Not policies specific to children, but more general policies and also different expectations that mostly prevent the issue from coming up. But I don't know what other South Korean universities do and anyway, your university is your university: it is (I suppose!) allowed to do things its own way. If you do not find written regulations of the sort I mentioned above, I would talk to your colleagues -- and especially, to tenured faculty; I also see that you are a master's student, which also may be relevant in terms of how much you are permitted to rock the boat -- and find out how they deal with the situation. If several other faculty members have successful "no children in the classroom" policies, then you should be able to implement yours. If you are the only one you know in your university who wants a "no children in the classroom" policy: because you are a graduate student instructor, I would advise against pursuing that.
Further Added: Please read the comments below about "drop ins". The policy I describe above is very standard in the United States. It seems that in certain European universities the culture (and perhaps regulations) are quite different.
Being a parent and a college student is tough. Not everybody has good access to childcare and even if they do, things happen. Surely the mere presence of a child in class can't be much of a distraction except for a few moments at the start of class. If the child is quiet and well-behaved, why not allow it? (Your policy could say that distractions, including noisy children, are not allowed.) It'll make some of your students' lives just a bit easier.