I really wish people would stop saying this. Even if you don't mean to trivialize what blackface actually was, that is basically the net effect. It's a terrible analogy on pretty much every level.
When the analogy is a worse thing than what is being likened, it is not trivialized. The thing being likened is moved to the level of the analogy and compared to see if the analogy is apt.
For example, "my teacher is a Nazi" does not make anyone think Nazis are only as bad as teachers, only that the student is overreacting (comparing the teacher to something that is on another level, i.e., worse.) The inverse, "Nazis were basically strict teachers," is a horrible thing to say as it brings Nazis down to the level of something benign.
To say The Big Bang Theory is blackface does not make a person wonder if blacks were actually only mistreated as much as nerds. It makes a person wonder if nerds are systemically exploited and mocked by social superiors. The answer is that in a very limited way, this happens, but it happens in an extremely irrelevant part of life (school) and nerds go on to run the world and are unquestionably full equals, possibly superiors, of more athletic people socially and politically. At no point here do we think blacks are only as mistreated as nerds.
On the other hand, saying something like "being black before the Civil Rights movement was like being the nerd in high school" is obviously offensive because we are bringing the black experience to the level of the analogy, being a nerd, and then making the comparison, and it's not even close.
They both set up caricatures of an unpopular group for the audience to laugh at. Like I said, blackface is far worse but I don't see how the analogy is "[terrible] on pretty much every level". It's much less hateful and damaging, certainly, but it's there in tone.
I was hoping to not have this debate that invariably comes up about how you can't compare anything to blackface because of how bad blackface was but I don't consider that a valid argument against the comparison.
I feel like the second paragraph wasn't there when I first replied to this, but if it was I just didn't notice it. I'd like to address it specifically either way.
I'm not at all saying "you can't compare anything to blackface." There are absolutely modern power structures that resemble it, and it is absolutely right to point them out and deal with them. I would never ever say that you shouldn't.
What I'm saying is that this is not a matter of scope but structure. Structurally, whatever you're seeing in Big Bang Theory does not resemble in any way what exists in traditional minstrelry. The people who are the butt of the joke in Big Bang Theory do not suffer systemic disadvantages in the way that Black people in the Jim Crowe era did, and they actually enjoy quite a lot of systemic advantages.
If merely caricaturing people for an audience to laugh at is sufficient justification for a comparison to minstrelry, then the entire comedy industry is and has always been guilty of it. As for unpopular, I'd also argue that there's a vast difference between unpopular and being treated as subhuman by law, to the point that they are not even close to the same thing. Black people being 'unpopular' is not why minstrelry was wrong.
Part of what makes blackface so horrible is the power imbalance that underlies it. In the days of minstrelry, black people were deeply and powerfully unequal in civil society. They had no recourse against the indignity done to them by blackface performance, and it was an indignity forced on them by a more powerful social class.
Try replacing nerds with black people in that paragraph and tell me it's at all comparable.
Where, in spite of that, your teachers were probably pushing you to perform, and when you got out you probably had no trouble getting into a good school which made it more likely for you to get a good job.
Kids are cruel, but their social order is not the real world. In the real world nerds are not a disadvantaged, let alone oppressed, group.
It is comparable. In the way that blackface allowed African Americans to work in venues and productions that they were otherwise prohibited, BBT allows nerds to be the subject of a mainstream sitcom.
BBT is not at all empowering the way that "Weird Science," for example, was, and highly-mainstream TV executive Chuck Lorre is of exactly a more powerful class who has allowed nerds to enter into a world where they were previously excluded. That BBT is only slighly less-awful than how nerds were previously portrayed is not a badge of honor.
How often does the show demonstrate dignified interactions between its main characters and the outside world?