His racial heritage was well known, and not particularly uncommon in France at the time, but there was undoubtedly casual and overt racism. His entire existence as we know it stems from one such incident.

In 1784, his father Alex, then a young well to do aristocrstic man took a white parisian woman on a date to a theater. A white officer present started making snide comments about how he must be the lady's servantn culminating in Alex Dumas being forced to kneel in front of the officer to ask for a "pardon", which he refused to do. A fight almost broke out and the police arrested both men. Nothing further came of it.

However, this event drove Alex to "prove" himself by joining the military and serving with distinction, rising to the rank of General. He had a rivalry with Napoleon during the time they both served, which led to Napoleon letting him rot in an italian prison for 2 years after being captured. Shortly after, he expelled black officers from the ranks throughout France and it's colonies.

The french were mostly "tolerant" of mixed race people. The elder Dumas was able to reach a high military rank, and was honored for service as any soldier might be. After Napoleon took power, that "equality" took a big hit, but by and large, biracial french people weren't discriminated against as heavily as in other places.

Dumas certainly was not hindered by it. He was the most prolific and popular author in France, and a good chunk of europe for much of his career. His heritage may have gotten him some snide comments and low level casual racism, but it clearly didn't affect his success in any meaningful way other than to drive him to excel further.