In a pot or large saucepan, melt the butter. Add the flour and whisk until smooth to make a roux. Cook until bubbly, then simmer gently 2 minutes.
Slowly whisk in the milk until smooth. Cook, stirring constantly, until warm.
Whisk in the mayonnaise, Dijon, and horseradish. Add the salt, pepper, and half the basil.
Add half of the cheese, stirring until mostly melted. Remove from the heat and set aside.