The down payment is typically (always ?) paid at/near the close of escrow anyway. Offering a large down payment at the end of escrow is essentially the same as offering a large down payment.
The deposit check is used as a sign of good faith. The larger it is the more the buyer puts at stake. There is always a risk that the buyer loses some of the deposit if they back out without cause (e.g. one of their contingencies).
The amount of deposit should depend on two things:
1. How much competition for the property
2. How much do you want to signal that you will follow through with the purchase. (strong buyer)
