Bigger signs the note "Red," and draws a hammer and sickle, the emblem of the Communist party. He does this in order to make the kidnapping scheme more believable, because he knows that there is widespread antagonism to communists: they are hated by the establishment and the authorities. "Reds'll do anything," he hears people say.
Or, alternatively, his intention might simply be to deflect suspicion away from himself by giving the police a convenient target—someone other than himself towards whom they will direct their investigations.
Bigger also knows that Jan is a communist and that Mary's family doesn't like him: to them, it would be more than credible that Jan—in his involvement with Mary—was simply using her for the Party's purposes. It's ironic that Bigger would try to place the blame on Jan, given that Jan has appeared to be sympathetic to him (although in a misguided, condescending way), but Bigger essentially trusts no one. He doesn't believe that Jan's attitude towards him is any more favorable than that of other people.
In the background of Bigger's extortion attempt are both the Loeb-Leopold murder (which had taken place in Chicago in the 1920s) and the kidnapping of the Lindbergh baby. In the latter crime, there had been a belief (alluded to in Native Son) that the Nazi party might have been responsible. Bigger's own attempt to blame a political organization for Mary's disappearance could have succeeded had it not been for the revelation of her remains in the furnace ashes and his obviously suspicious actions at that point in the story.