No, it wasn't. This often made claim is based on a passage of Phaedrus where Socrates is quoting an Egyptian legend, and the negative judgement of the invention of writing is done by a pharaoh inside that legend being told inside the dialogue.

Besides, the dialogue is not about writing itself, but about writing down political speeches and the consequences and impact of political speeches being read rather than heard.

Also, the dialogue dates from about three thousand years after writing was invented.

Some direct quotes from the dialogue:

SOCRATES: So, it is obvious to everyone that speech writing is not shameful, not in itself anyway (258d)

SOCRATES: We are left with the question of the appropriateness and inappropriateness of writing, and how it may be executed in a worthy or an inappropriate manner. Do you agree? (274b)

And this, a very relevant insight for an age when some are convinced that text analysis should suffice for intelligent work:

SOCRATES: Then a person who thinks he has left a skill in writing behind him, and anyone who, for his part, inherits this on the assumption that something clear and certain will emerge from writing, would be full of enormous silliness, and indeed ignorant of the prophetic words of Ammon in believing that written words are anything more than a reminder to a person who already knows whatever it is that the written words may refer to.

Ironically, the sharpest criticism Socrates made of writing in the often misread Phaedrus was of misreading texts.