There are at least three ways to make this argument.
First, you can argue that it is inherently Christian to sacrifice one’s self for others. When a woman becomes a prostitute to support her family, she is sacrificing herself to help them. In a sense, this is following the example of Jesus who sacrificed himself for the benefit of other people.
Second, you could argue that not all sins are created equal. You could say that allowing her children to starve or to otherwise suffer would truly be a greater sin than engaging in prostitution.
Finally, you can point to the example of Jesus and how he dealt with prostitutes. Jesus did not judge the prostitutes with whom he came into contact. Instead, he dealt with them with compassion. This does not mean that prostitution is acceptable, but it does mean that we should not be judgmental about those who become prostitutes.
I would never be able to argue this point because it would be against my conscience.
Firstly, if you were really a Christian, you would harmonize with other Christians, and those other Christians will help you support your family.
Secondly, God always provides. Becoming a prostitute shows lack of trust in God's provinding hand. And this isn't a load a baloney either, because it has happened many a time. For example, one year the church needed extra money for one particular reason. My parents were already tight for money, but still put extra money forward. Things couldn't be any tighter. Anyway, the next week, my dad got a payrise. Coincidence? I don't think so.