An alcohol problem in one's past does not preclude holding a first class medical. The nature of the problem, and its recency, and its current status are considerations. Item 18.0 on the application asks if you have ever been diagnosed with, or presently have alcohol dependence or abuse. You are able to provide detail thereafter. The FAA is chiefly interested in how alcohol impacts your health, judgement, etc, moving forward, as a pilot. There's no such thing as an ex-alcoholic, so present trends and condition and use are important. It's one thing if you were arrested for a DUI yesterday, vs having not touched a drop in twelve years, with regular AA attendance, etc. In general, the FAA looks at the prior two years, and also on a case-by-case basis.
14 CFR 67.107(b) defines substance abuse, with regard to the FAA medical application:
§ 67.107 Mental.
Mental standards for a first-class airman medical certificate are:
(b) No substance abuse within the preceding 2 years defined as:
(1) Use of a substance in a situation in which that use was physically hazardous, if there has been at any other time an instance of the use of a substance also in a situation in which that use was physically hazardous;
(2) A verified positive drug test result, an alcohol test result of 0.04 or greater alcohol concentration, or a refusal to submit to a drug or alcohol test required by the U.S. Department of Transportation or an agency of the U.S. Department of Transportation; or
(3) Misuse of a substance that the Federal Air Surgeon, based on case history and appropriate, qualified medical judgment relating to the substance involved, finds—
(i) Makes the person unable to safely perform the duties or exercise the privileges of the airman certificate applied for or held; or
(ii) May reasonably be expected, for the maximum duration of the airman medical certificate applied for or held, to make the person unable to perform those duties or exercise those privileges.
Verification that you can hold a first class medical is often recommended as the first step in pursuing flight training. That's not normally my recommendation, but certainly before you invest a lot of money or entertain a career change, you should ensure the ability to hold the medical; without that, one can't work for an airline. One can work as a commercial pilot doing many other things, with a second class, but there's not much difference in the exams.
In short, you're not precluded. Pilots working as pilots have alcohol problems, and still come back. It can certainly hinder one's career and depending on the recency and the degree of the problem, employers can look on it unfavorably, but alcohol problems can be overcome. If you're an alcoholic now, then you shouldn't be flying. If you've had alcohol problems in the past, but these are not part of your world now, then first of all, congratulations, you've over come a lot, and second of all, they don't preclude you from becoming a pilot.
The first thing I suggest that people do, if they're looking at flying for a living, is take a discovery flight. Most schools offer them. Get an hour of instruction or an introduction to flying, see what you think. If you're making this decision with a spouse or significant other, it doesn't hurt to have them go with you. If you do take your wife, make sure it's a morning flight on a calm day, and do a little sight seeing. Pictures of the home, local stuff, etc. Positive experience. But don't neglect the purpose of the flight, which is to see if it's for you. Chances are, if you feel the drive to do it, then it is for you. The hardest part of learning to fly is paying for it. Find a way past that, and it's not a bad gig.