Praise be to Allaah.

This is something that definitely needs to be taken up in the courts of law. One should not remain with an alcoholic as he harms his wife and his children. One should remain away from him unless Allah guides him and he returns to what is correct. If the judge separates the two, usually, the children are given over to the mother as she is most qualified for them while he is not qualified. As long as he has the problem of being an alcoholic, he is not suited to bring up his children as he will destroy and ruin them. Therefore, she has more right to her children than him, even if they be boys. This is what is usually done by the judges and this is what is obligatory. The children must be with her because she is better than him and he is an evildoer. If she refuses to go back to him, she has done well, as such a living condition is harmful and dangerous for her. If he also does not pray, then it is obligatory not to return to him for the one who abandons the prayer is a disbeliever-- and protection is sought from Allah. The Prophet (peace be upon him) said, "The covenant between us and them is the prayer. Whoever abandons it has committed disbelief."l
It is not obligatory to remain with the one who does not pray.2
"They are not lawful [wives] for the disbelievers nor are the disbelievers lawful [husbands] for them" (al-Mumtahina 10).
[That is,] until Allah guides them and they repent. The woman should go to her family or stay with her children and not let her husband come to them until he repents to Allah and returns to what is right.If he prays but drinks alcohol, then that is a great sin and great crime.
However, he is not a disbeliever but an evildoer.The woman has the right to prevent him from her and to leave him. She is excused for that act. If she remains patient and has the ability to be patient, there is no harm in that option either.

Shaikh ibn Baz

Footnote
1. Recorded by Ahmad, al-Nasal, al-Tirmidhi and ibn Majah. Al-Albani calls it sahih. Al-Albani, Sahih al-Jami, vol. 2, p. 760 .
2. This sentence, as is clear from the remainder of the response, must not be misunderstood. It does not mean that one may stay with one who does not pray if one wishes to do so. It is not an option to remain with a husband who has been deemed a disbeliever.