How do you prove mother unfit in Missouri?

You don't. You provide proof of the Best Interest of the child. Targetting the mother is never a good idea. See links

The factors used to determine that a parent is unfit are generally governed by state laws with child endangerment being the determining factor. The following include some of the reasons a parent may...

Leave her with a child that has been promised a basket of chocolates if he is naughty.

A cheating wife is not necessarily an unfit mother, unless she is not looking after the children properly and that fact isn't only the purview of a cheating wife.

I would think you would need to prove it in court. Whatever area you think they are unfit, you will need physical proof, not just hearsay, for example: drug addiction, mental illness, sexual abuse,...