The more you put down, the less your payments will be. If you only have a small down payment, your credit score will determine what a dealer will accept.
This is typically dependent on several factors. The amount that you will qualify for may change considering your credit score, the "out the door" cost of the vehicle, and even the dealership you're dealing with. If you find a dealer that provides in house financing you may have better options.
Each car dealership and bank has a different policy, but if you can only afford a small down payment then try and go to one of the dealerships that advertises that they can finance anyone. They really can't finance anyone, but they are more likely to accept you than one with stricter policies would.
If you have less down payment in your hand you need to be invest at least ten to twenty percent of the down payment as and when you need to buy a car of your own. If you pay a considerable amount your co signer would have a good credit in the car mode.