If only her husband cooperated, she could probably eliminate most of the household worries.
Literal
Husband [subject-が] cooperate-do-for-her [even just-さえ] [if-すれば], she [topic-は] home [from-から] majority worry seeds [object-を] remove thing [subject-が] can-do probably.
~てくれさえすれば = 'if only he would do X for her.' さえ~ば sets the minimum condition. 悩みの種 = 'seeds of worry / sources of trouble.' 取り除く = 'to remove/eliminate.'