Just like anyone who applies for a job and has their background checked. They can find out about any arrests and not just DUIs. They can also do credit checks to find out your credit history. It can become a bigger problem if you have a prior dui and don't divulge it in your application, if asked.
If I were you, I would make an anonymous call to the licensing board for more information. Only the licensing board can tell you how they perform background checks and whether DUIs will prevent a person from being licensed as a nurse.
