。
Her one and only concern is the safety of her children.
Literal
She [subject-が] [just-ただ] one [worry-気がかり] [the thing-なのは] children [genitive-の] safety is.
A cleft construction stretches the spotlight onto 子供たちの安全: 'as for what (she is the) one thing concerned about, [it] is the children's safety.' Reading the parts: 彼女がただ一つ気がかりなの ('her one and only concern,' literally 'the thing that is her one concern') — the が marks 彼女 as the subject of 気がかり (a な-adjective meaning 'concerning, worrying'); の nominalizes the whole phrase; は contrasts and topicalizes; the predicate after は names what it is. This のは...だ frame is the workhorse of Japanese clefts ('what X is, is Y'). The sentence reads with formal weight, fitting a context like a profile or an interview.