We would normally say "behind 'on' her rent" here. Imagine you had to pay your rent on a monthly basis, and by the end of the sixth month of the year you had only paid 3 months rent. Then you would be "behind on your rent." The person saying "if she's behind on her rent, I don't know what she'll do" doesn't know whether the woman in question is up-to-date with her rent payments, but predicts an impossible situation for her if she isn't.

She has not made some rent payments.