Personally, it irks me to no end.
I don't mind the casual supporter for logical reasons "Defend the canal ward gate, etc". Those make sense.
When 1 or 2 people move for equity, that's fine as well.
But when PC's come en legion to support the rights of a well known, evil, sneaky race, it drives me up the wall.
If you support Goblins, why not Drow? Or tieflings? Or Anything for that matter?
Goblins and Kobolds are renown for their trickery and evil. Much like Drow are known for their malice and evil. Or tieflings and their shadiness and evil.
Defending them on the platform of "Well they are living things too!" doesn't matter. They are 2nd if not 3rd class citizens because of what they are. The 1960's-70's didn't occur in FR. No one has marshaled an army of men to fight for their rights, and in my mind canonically they wouldn't for the fact that things -are- so black and white in this world. Gods -do- exist. Evil and good -does- exist. A good drow occurs once every 3rd blue moon, yes, as does good creatures of other races, but my issue is this:
"A bad apple spoils the entire bunch" does not really apply in FR. One bad aasimar would not make you say "AASIMAR ARE EVIL MONSTERS HELLBENT ON DESTRUCTION!". The converse should apply as well. One good character is not redeeming enough for the entire race. Goblins are evil, and will kill you/lie to you/whatever. Just because a character met one who didn't, or a PC who faced racial injustice thinks they can fix it (A common act in many places), does not mean that you should campaign for equality. Majority of the time, I'd expect a character who -has- face racial injustice to perform even worse against lesser/more hated groups, because that's just human nature.
Faerun and the forgotten realms usually in general has a dirty caste/feudal system. Goblins and Kobolds are at the bottom of the barrel, and the facts are there that it pretty much will never change. On the surface this is well known usually, and in Underdark I expect it would be even moreso known since it is even more cuthroat and "Chaotic" in general. People would worry of their own hides and agendas, rather than some smelly goblin tribe hiding in the back alleys causing trouble.
I also feel that goblin/drow/illithid/whatever sympathizers should and will be met with hatred, and should be expecting that.