If I seek help for my alcoholism, could that affect my chances of applying for citizenship?

I applaud your intention to obtain professional assistance to address your alcohol use, and I wish you much success. The USCIS will not have access to your medical records, but it does have access to DUI and other police records, divorce decrees and some other records that could reveal a problem with alcohol, and additionally one of the questions on the naturalization application to which you must accurately reply is "Have you ever been a habitual drunkard?" (I know that is an awkward reference to alcohol problems, but there are historical reasons for that language). The USCIS considers being a habitual drunkard to mean an applicant for naturalization lacks good moral character, but generally the good moral character issues is limited to the 5-year period associated with naturalization eligibility (3-year period for many people married to U.S. citizens). "Habitual drunkard" is a somewhat vague term, and generally it refers to a person given to brevity or excessive use of alcohol and has lost the power or the will to control his appetite for it. A clinical medical diagnosis of alcoholism would satisfy the definition, but a general concern by a person that he was drinking more than he would like probably would not, by itself, satisfy the definition.