We don't serve their kind here.

1. Organic beings may not want to mix with machines.
2. What exactly do droids eat and drink? Nothing. Bad customers.
3. Someone has to be the underclass, and in Star Wars, it's droids.

White privilege. Thats the only explanation

It was a racist scene. Maybe they'll cut it from future releases.

they posted about Nazis

C3PO had a line in either the film or the book about not needing lubrication, so among other issues the bar didn't want to lay in a stock of lubricant. Or maybe they'd had trouble with the tough guys who came to the cantina bringing their battlebots along.

But really, in filmic terms it was meant to establish that this was a rough-as-fuck place run by a complete asshole, one who didn't give a rat's ass if his patrons killed each other (as long as the dead being wasn't running a tab), and one who was deliberately offensive to establish his toughness. It also required the Droids to go out and scout the town, which gave the audience more information about the Imperial presence in a few seconds of film.

This is just a theory, but films and other stories always had a long history of robot hate since the very idea of automation lead to less jobs. Given how poor Tatooine seems to be, I don't think it's too surprising they'd have some prejudice towards droids.

With all the shady stuff going on in those places, bringing a droid in may keep people away for fear of being recorded / identified for doing said things.

"We don't serve their kind here", simply because they cannot. Therefore, they only serve people who can drink and eat. Bad for business otherwise. Taking up space for no reason at the Cantina.