Yes, more down will help. No question. For example, I also have minimal credit which HURTS your over all credit score. 3 months ago my wife and I bought an 08 Honda Odyssey. We have no credit cards and no loans out on ANYTHING. We pay cash for everything and have been for years. BUT, we want to buy a house in the next 5 years so we NEED credit to do that (assuming we finance). So how to you get your credit score up? By having loans/credit cards in good standing.
That said, our credit scores were both around 500. BAD credit in the eyes of a bank but not because we don't pay our bills (we have none), because the bank and credit reporting agencies see no proof we can pay debt month to month. Soooo... How do you get a loan with a "bad" credit score? You put mad cash down! Our Odyssey was listed at $19,800. We put down $8,000 cash and walla... We got a car loan. 28 days later both our credit scores shot up 40 points! And the best part was the bank we financed with gave us a credit card with them because we had an auto loan with them. We use it for food/gas only and NEVER use more than 20% of it's value and ALWAYS pay it off on the due date (Never before because the balance has to be reported or you'll actually hurt your credit from growing). Also, another bank of ours does a secured credit card which uses your own money as the credit balance AND reports to the credit agency. This is AWESOME for building credit because it's already your money and it's working to build credit. Nothing to lose!
In about a year, we should be closer to the 700 mark which is pretty good. After 2 years, we should be in the 800-850 range which will give you the best financing you can get in the housing market and with just a simple couple credit cards we hardly use but keep a balance on. EASY!
EDIT: Something to note here is that your debt to income percentage plays a HUGE part in how much you can be approved for. It's a real balancing act so always take caution when starting to build credit or your credit score will go down, not up. I know this because I used to have a credit score in the 800 range about 10 years ago. Once I went cash only and was debt free, my credit score tanked. Debt free is nice, but there are times when it can really burn you. Like if you want to buy a big ass house or a jet..... or both