I am a few years away from being done my PhD in philosophy, and plan to seek work in the field. I have a DUI on my record from when I was 18 years old. Will anyone care?
Nobody will care.
A drug-related crime (of which a DUI is one example) from your teenage years is just not at all relevant to your professional qualifications and your ability to do your job. If it were more recent, or if it were a more relevant crime (e.g., a violent offense), then it might be a different story.
As noted in the comments, though, don't try to hide it if somebody asks about it. You don't need to bring it up, but if somebody else does, just tell the honest truth and you won't raise any red flags.
Oh, and in the future make sure you have some sympathy for the students who are making the same mistakes that you did.