To find the cost per player, first calculate the total amount collected from the players. Each player pays $5, and there are 60 players.
Total amount collected from players: \[ 5 \text{ dollars/player} \times 60 \text{ players} = 300 \text{ dollars} \]
Now, to find the amount that the league has to pay, subtract the total amount collected from the total cost of the banquet: \[ 960 \text{ dollars} - 300 \text{ dollars} = 660 \text{ dollars} \]
Now, to find the total cost per player, we need to calculate the total cost per player, which is the total cost of the banquet divided by the number of players: \[ \text{Total cost per player} = \frac{960 \text{ dollars}}{60 \text{ players}} = 16 \text{ dollars/player} \]
Therefore, the cost per player is: \[ \boxed{16} \]