But one of my partners feels there shouldn't be a line at all. The Armed Services certainly feel that way, and there are many aphorisms that seem to make the case. The Army's case for separating the officers from the enlisted was that in battle you needed to make decisions about sending people into dangerous situations without knowledge of personal information that might make you hesitate to send the most appropriate person or group into a life-threatening situation. I assume this isn't your situation. Certainly there are some problems that can and do result from close relationships at work.
Managers sometimes show favoritism to their buddies.
Some result in improper sexual conduct. Some result in employees taking advantage of a relationship with a supervisor to cover their poor performance. Having a policy or discouraging personal relationships won't eliminate them entirely. In my experience, such policies only make employees angry with the company's management. Since they don't work effectively and tend to decrease morale, such sanctions don't seem like an effective way to tackle the situation.