It's probably safe to say that if there's one place where minorities would have been glad NOT to be included, it would be in the population of the Bastille. And in fact the great majority of the castle's inmates were white, European and male. Most were French, but some were English, Irish, German, Italian, etc.
Constantin de Renneville, one of the first to write a detailed account of his imprisonment, wrote:
During the time of my imprisonment (1702-1713), I saw at the Bastille, not only French, German, English, Scottish, Irish, Spanish, Polish, Swedish, Danish, Muscovite people and generally people from every nation in Europe, but what is more people from all parts of the earth, even the farthest removed and hardly known. I saw there Africans, Asians, Americans, Turks, Moors, Greek popes, dervishes, prelates, ministers and monks of every color, I saw princes, such as the son of the king of Morocco, who was treated in a barbarous and cruel manner...
...I have seen every sort of person and nation confounded one with the other in the hard brass press of a tyrant who moved all the springs of his infernal machine to extract the substance from his unhappy pigeons...
Unfortunately, he gives few specific examples and neither the published archives nor other sources mention any "Africans, Asians, Americans" etc. Which is not to say they did not exist, only that that part of Renneville's statement seems not to have so far been confirmed by others.
However, one black inmate - probably African, though he might have been born in Europe - can be positively identified. Bastille records from October and November 1781 list the residents of No 1 in the Treasury (Trésor) Tower as "Saint-Lubin and his negro"; that is, Pallebot de Saint-Lubin and Narcissus (Narcisse), entered January 17, 1781, transferred to Charenton, November 19, 1782. It was not unusual for servants to be imprisoned with their masters, which appears to be the case here. Narcissus' "loyalty" here might of course have been constrained. However, when Saint-Lubin later escaped from Charenton, he did take Narcissus with him, so there may have been a real bond between the two men.
A more tantalizing case is that of the councilors of Saint Domingue (now Haiti), who were sent from Port-Au-Prince all the way to Paris just to be imprisoned in the Bastille. They were arrested for resisting the reestablishment of local militias, which had been divided by race: white, mulatto, free black. While it is likely that the councilors - all men of privilege - at least looked white, it is not impossible that some were of mixed race. In his paper The Free Colored Elite of Saint-Domingue: The Case of Julien Raimond, 1744-1801, John D. Garrigus writes:
In 1731 an official inspecting the militia in [one] parish informed the governor that "there are few whites of pure blood there because all the whites willingly ally themselves by marriage with the blacks, for they, by their thrift, acquire property more easily than the whites." Yet the Bainet census of 1730 counted 317 whites and only 12 free non-whites, according to Moreau de Saint Méry. Those economically-secure free coloreds whose presence so impressed the militia inspector in 1731, men and women who despite their color had never been enslaved, seem to have been counted as 'white' by the local officials who submitted the census reports.
Further research into the council's make-up and into the genealogies of the men in question, might establish that one or more were of mixed race. For now, however, it seems that Narcissus retains the (ambivalent) distinction of being the one documented inmate of color in the Bastille.