I am in engaged and my fiance' had a misdemeanor DUI will that affect our ability to rent an apartment?

Difficult to say but, probably not. The landlord is probably more interested in your credit rating. In the event they ask for a copy of your criminal record then, keeping in mind the non-discrimination laws that exist almost everywhere, the decision would have to be made by them. Frankly I wouldn't worry about it , but I wouldn't volunteer the information either.