A consular officer can deny your visa application for any number of reasons, such as visa preference problems or the presence of a criminal history.
A DUI conviction will most certainly have an effect on your green card application. The extent of that effect, however, will depend on several factors.
If you are petitioning USCIS for a green card, you must attend (and pass) a medical exam proctored by an approved physician or doctor.
If your fiancé’s immigration application has been denied, they may be able to apply for an I-601 Application for Waiver of Grounds of Inadmissibility.
The U.S. Department of State has indicated that the most common elements of a crime involving moral turpitude are (1) fraud; (2) larceny; and (3) intent to harm a person or property.
Being found inadmissible is not a death sentence to your immigration process. However, some grounds of inadmissibility are more difficult to overcome than others.
While you can get a fiancé visa prior to being married, you may only apply for a green card after you finalize your marriage.