Melt butter in a medium saucepan set over medium heat. Whisk in flour, one tablespoon at a time, until dissolved into the butter. Gradually pour in the milk, whisking constantly until the mixture is creamy and smooth. Continue to whisk the mixture as it cooks and thickens.
Add the cheese, 1/2 cup at a time, whisking constantly, until the cheese has melted completely. Stir in the salsa and the sausage. Serve warm. You can also reduce the heat to a low simmer to keep the dip heated for serving or transfer to a slow cooker on a warm setting to serve warm for a party.