Her husband drinks alcohol; should she stay with him?

Firstly: we ask Allaah to help your sister to deal with this
calamity and to bring her relief, and to guide her husband to give up this
major sin in which he is persisting. 

With regard to the ruling on her staying with him, it is
permissible for her to do so if she is certain that she and her children
will be safe. 

Shaykh Ibn ‘Uthaymeen
(may Allaah have mercy on him) said: I think that if she has advised him and
he has not responded, then she has the right to ask for an annulment of the
marriage. But there may be reasons why she cannot annul the marriage,
because she has children and annulment may cause further problems. If his
sin does not reach the level of kufr, then there is no sin on her if she
stays with him for fear of evil consequences. But if his sin reaches the
level of kufr, such as if he does not pray, then she should not stay with
him a second longer. 

Al-Liqa’ al-Maftooh, Q. 518. 

Secondly: Our advice to her is that she should weigh up what
is in her best interests, and consult her family and relatives, who know her
situation best. Then she should ask Allaah’s guidance (by praying
istikhaarah), for Allaah will never let her down. As the report says: No one
loses by praying istikhaarah and no one regrets consulting others. Whether
she chooses to annul the marriage or to stay, let her accept what Allaah has
decreed for her and be patient and seek reward. Allaah says (interpretation
of the meaning): 

“but give glad tidings to As‑ Saabiroon (the patient)”

[al-Baqarah 2:155]

And Allaah knows best.