。
She has a heavy-drinking husband.
Literal
She [topic-は] heavy-drinking [genitive-の] husband [object-を] is-having.
大酒のみ ('heavy drinker') is built from 大酒 ('heavy drinking, drinking a lot') + のみ (the ます-stem of 飲む 'drink') used as a noun. The pattern [object]+[verb-stem] gives an agentive noun — like 力持ち ('strong-armed person, lit. 'force-holder'), 物知り ('know-it-all, lit. 'thing-knower'). Note the slightly unusual phrasing 夫を持っている ('hold a husband'); in everyday speech 夫がいる ('have a husband') would be more natural. 持つ for 'have a spouse / family member' is a slightly stiffer or more literary choice.