This quote is found in Chapter I of The Souls of Black Folk. It is part of a passage where du Bois is lamenting the fact that much of white prejudice against blacks is unfounded. He says that some people think that whites are prejudiced against blacks because whites are a higher, more civilized race. Du Bois says that blacks will acquiesce to however much prejudice is caused by whites being more civilized than they are. However, he says, much of racial prejudice is not justified. It is caused merely by hatred. This is where the line you quote comes in.
To understand what the line means, you have to understand the words used. If something is “pervasive” it is widespread. It is everywhere. So something that is “all-pervading” is something that is found everywhere and in everything. To “inculcate” something is to teach it. This is usually used to refer to attitudes more than facts. We “inculcate” moral values, but we do not usually say that a teacher “inculcates” your multiplication facts. Finally, if you “disdain” someone or feel “disdain” for them you look down on them. So, what your line means is “a desire that can be found everywhere to teach people to look down on everything black.”
This, du Bois says, is a kind of prejudice that African Americans will not accept.