This video highlights some of the ways alcoholism can affect children. One of the ways mentioned in the video is that children will lose a lot of trust and/or have a hard time trusting other people. Alcoholic parents will not be able to keep their word a lot of the time and as such the bond between parent and child will weaken. 
In the case of Treeless Mountain, while Jin and Bin's mother is not seen to be an alcoholic, the substitute parental figure that is Big Aunt serves this role. Because Big Aunt drinks often, she neglects Jin and her sister often as well, always being either passed out or too hungover to provide for them. 
The big takeaway from this has more to do with a friendly neighbor of Big Aunt's, a woman with a disabled who is very kind and friendly to Jin and her sister. Although Bin is happy with her new friends, her sister  Jin isn't too happy, Jin always tries to go home early and wants to interact with the woman and her son as little as possible. This is because in Jin's mind, she believes the woman to be "replacing" her mother in terms of a parental figure. She sees the woman's kindness as as such and does not want it, Jin believes no one can replace her mother and this is seen through her emotions and actions. Obviously we as the audience know that the woman truly is a kind soul and only trying to help but little Jin cannot comprehend this and so she tries to create some distance between them. Jin experiences firsthand the negligence of her aunt and because of her alcoholism, Jin can't see her as a good substitute for her mother, leading to distrust of any other would-be parent figures.

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