My salaried employee requests an unpaid day off. I am happy to grant it, and to reduce the weekly salary by the one day. Is that appropriate?
I see this is in the United States. Is this a FLSA-exempt employee? It's important, because the Salary Basis Test stipulates that they are entitled to their full salary for any week in which any work is performed. Even if he asks for the unpaid day in writing, an employee cannot waive the protections of the FLSA.
Granting the request could open a can of worms for you, and you should consult with a labor attorney before allowing such requests. Now, if you just let them take the day off paid, no can of worms has been opened. Frankly, an occasional paid day off is kind of a perk of being salaried vs hourly... you know, given the expectations of unpaid overtime.
I assume by asking the question, you don't have a HR department. I recommend that you contact a local contract HR company, and pay them to handle similar issues.
You should also consider the cost of this one day. If you give them the day off, and then jump through all the hoops to make sure you don't pay them for that day, you'll spend (probably) more money than if you just paid them. Plus, if you just give them the day as a paid holiday, you don't need to do anything on your payroll. Simple for you, and gets an enormous amount of goodwill from the employee.