The Greybeards claim this when you mention you're working with the Blades. In retrospect, it seems a clear case of foreshadowing their eventual betrayal over the issue of Paarthurnax. However, it seems clear he's speaking from long and vast experience. The Greybeards LOATHE the Blades and vice versa, so there's some definite history.

What do you think it is?

The Greybeards don't approve of the Blades' methodology. One group is about peace and contemplation, the other is about violence and subterfuge. I also wouldn't doubt the current Greybeards know about the Blades' previous attempts to kill their master, which obviously wouldn't sit well with them.

The Blades don't hate the Greybeards, they simply have no respect for them or their ways. They think of them as old fools for sitting in their monastery in peaceful contemplation when they have the power to shape the world essentially however they please. It's only when they find out that the Greybeards are protecting Paarthurnax that they begin to consider them an enemy.

On the other hand, the Greybeards hate the Blades because they see them as reckless, arrogant, and power-hungry. They nearly drove the dov extinct for the sake of the vendetta of a group of vampire snake-men, when Paarthurnax had proven that they could control their impulse for domination and become valuable allies and friends, and then Esbern and Delphine picked up right where they left off. They tried to lead the Dovahkiin astray from the Way of the Voice, and reshape him into a weapon they can use for their own agenda.

That said, I don't believe Arngeir when he says the Blades have never served the Dragonborn. The Blades of the Third Era were obviously fiercely loyal to the Septims. By the reign of Uriel Septim, I think they had forgotten much of their original mission and had instead embraced their role as bodyguards, spies, and agents of change.

Pretty much. They care more about their stupid obsession with killing dragons than either serving the Dragonborn or combating the group of mer that nearly wiped them out. Shame, since they're otherwise one of the coolest organizations out there.