1. Mrs. Robinson has three children to support and she doesn’t make very much money at her job. She is barely keeping her head above water
2. Peter is having a difficult time at the university because he wasn’t very well prepared academically, but he is somehow managing to keep his head above water.

Antonym: in over (one’s) head. Compare to: make ends meet; get by. Keep one’s head above water and make ends meet mean having just enough money but no extra, although the former conveys a greater feeling of desperation. Keep one’s head above water can mean survival in a financial or other sense, whereas make ends meet always refers to a financial situation.