"I have to babysit my little sister" means that you will be looking after her while your parents are away. "I have to babysit with my little sister" means that you and your little sister will be looking after some other children (the sentence doesn't specify who those children are), while their parents are away.

If you are going to be caring for your little sister, you do not need to use "with". It's sufficient to say, "I have to babysit my little sister".

'I have to babysit with my little sister' means that you and your sister are caring for another person. 'I have to babysit my little sister' means that you will be caring for your little sister.

My view: (native speaker, New Zealand) Most natural, as below answers: "I have to babysit my little sister" "I have to babysit with my little sister" <-- ambiguous, could be with, or could be looking after her. Probably taken to be looking after her, as it's more likely