Here's the thing: all this affirmative action-y, treat blacks special-y stuff was.... at least needed at the time period. Not only were black people not treated well by whites, it was practically law for that to happen pre-1960s. (See also, Jim Crow laws). Slavery was never meant to be constitutional, but evil people decided that owning people was well and good. The 13th amendment ensured this, basically by framing blacks as men and not chattel.
With Black History month (and its predecessor, Negro History Week), it does help highlight actual achievements made by black people, and again, it was necessary for the time coming off the heels of the abolishment of slavery and sliding in before the Civil Rights movement of the 60s. It was helpful in boosting morale, and one could argue that it still does to some degree, but this was unbelievably useful when success stories told were mostly about white people. Nowadays, it's hardly necessary.
With blacks achieving things that we never had prior to the Civil Rights movement, it's almost pointless to have these kinds of months. I dare say that nowadays it could be racist. The whole point of Civil Rights is to treat us as equals, and while we may not be anywhere near perfect (yes, racists still exist, I'm not denying that), we've come so far along that we really don't need to point out our achievements by the levels of melanin in our skin. Not to mention, we're all becoming more and more mixed as a society, that the differences in ethnicities is get ridiculously pointless. So to separate an entire month for just black people could easily be racist from multiple angles.
1) "Oh, you blacks have come a long way, but you're still only diminished to one month of a celebration for your vast culture and achievements, while whites can be celebrated all year long! Let's highlight a few people who we deem 'acceptable' for black people."
or
2) "Blacks deserve their own whole month based on the color of their skin! Whites... ehh not so much. They're not special enough."
or
3) "While there have been leaps and bounds in achievement among black people, and an overall rise in general ethnic acceptance, let's just have specific months for specific people. Keeps racial tension up."
1/2