How do I, or should I, bring up research that I did under someone who was fired for another reason? I worked under a professor whose research was funded by a non-profit organization. They were released from the university for some inappropriate behavior. The research that they have been publishing has been solid so, were it not for this incident, I would happily put it on a resume or application. The incident does not involve plagiarism or really anything that would make someone suspicious of the validity of their findings or published works. I understand that this is vague, but it is intentionally so for privacy and security reasons. 

Until now, I have been putting down the name of the research program that the professor led that I was a part of. However, I am worried that employers may research the program, the professor, and the incident that resulted in their termination and make judgments about my character and work. 

I am trying to finish the work that I started either alone, or with a professor at another university. I plan on acknowledging the university that I am enrolled at and the non-profit for providing the equipment and grant money, respectively.  If I am able to continue the work with another professor at another university or simply publish on my own, then I may just discuss that aspect of the process in applications and interviews, but until the work is completed, any advice on how I should proceed?
From the tags, it seems you are an undergraduate. Therefore, I am assuming you don’t have a lot of other research experience to list. You also specify that the misconduct was not research-related. 

Based on these factors, I would suggest keeping this work on your resume/CV. I would stress the topic and content of the work rather than the person/group. Perhaps where you would normally list the faculty you worked under, you could list the university and department. Or you could perhaps list a post-doc who was a more direct supervisor, if applicable.

You are concerned about negative consequences of being affiliated with the professor in question. People don’t generally hold undergraduates responsible for professors’ conduct, so I think anyone who really considers the issue will not blame you. Personally, I would be more inclined to pity an undergrad in such a situation.