"How much does a driver cost for a day?"
In Chinese, we usually put the subject before "how much", so it becomes "The cost （is） how much? 【费用是多少】" and not "How much is the cost?".
一天的费用 is "【one day's】 cost" or "cost 【of/for】 one day", that's the meaning of the “的” here.
司机一天的费用 = The cost of the driver for one day. （I cannot explain why we don't say 司机的一天的费用...I can only say that the first 的 is not necessary but the 的 in 一天的费用 is a must.）
The whole sentence: The cost of the driver for one day is how much?