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.