Do you have to pay to access the Cascades du Moulin?

No: access to the Cascate del Mulino is free, and you don't need to book. Don't confuse it with the Parco Termale of Terme di Saturnia, which is a private establishment with paid entry and landscaped pools. At the waterfalls, you swim in the natural pools; there's no ticket to buy to access them.

The budget to plan for is mainly parking: the nearby parking lot is paid and the observed rates are around €2.50 per hour, but check the price displayed on the parking meter on the day of your visit. It is located a few hundred meters from the pools, allow about 5 minutes on foot. Showers, lockers or other services on site may also be paid. So bring something to dry off and store your belongings, and don't leave anything valuable unattended. In short: free swimming, but an outing that can still involve some costs if you come by car or use the facilities.