Before doing anything else: you want a lawyer involved right from the beginning, to make sure that something reasonable happens with the house if one of you dies or leaves. Seriously, you'll both be safer and happier if it's all explicit.
How much you should put on the house is not the right question. Houses don't sell instantly, and while you can access some of their stored value by borrowing against them that too can take some time to arrange. You need to have enough operating capital for normal finances, plus an emergency reserve to cover unexpectedly being out of work or sudden medical expenses. There are suggestions for how much that should be in answers to other questions.
After that, the question is whether you should really be buying a house at all. It isn't always a better option than renting and (again as discussed in answers to other questions) there are ongoing costs in time and upkeep and taxes and insurance. If you're just thinking about the financials, it may be better to continue to rent and to invest the savings in the market. The time to buy a house is when you have the money and a reliable income, plan not to move for at least five years, really want the advantages of more elbow room and the freedom to alter the place to suit your needs (which will absorb more money)...
As far as how much to put down vs. finance: you really want a down payment of at least 20%. Anything less than that, and the bank will insist you pay for mortgage insurance, which is a significant expense. Whether you want to pay more than that out of your savings depends on how low an interest rate you can get (this is a good time in that regard) versus how much return you are getting on your investments, combined with how long you want the mortgage to run and how large a mortgage payment you're comfortable committing to. If you've got a good investment plan in progress and can get a mortgage which charges a lower interest rate than your investments can reasonably be expected to pay you, putting less down and taking a larger mortgage is one of the safer forms of leveraged investing... IF you're comfortable with that. If the larger mortgage hanging over you is going to make you uncomfortable, this might not be a good answer for you. It's a judgement call.
I waited until i'd been in out of school about 25 years before I was ready to buy a house. Since i'd been careful with my money over that time, I had enough in investments that I could have bought the house for cash. Or I could have gone the other way and financed 80% of it for maximum leverage. I decided that what I was comfortable with was financing 50%. You'll have to work thru the numbers and decide what you are comfortable with.
But I say again, if buying shared property you need a lawyer involved. It may be absolutely the right thing to do ... but you want to make sure everything is fully spelled out... and you'll also want appropriate terms written into your wills.
(Being married would carry some automatic assumptions about joint ownership and survivor rights... but even then it's safer to make it all explicit.)
Edit: Yes, making a larger down payment may let you negotiate a lower interest rate on the loan. You'll have to find out what each bank is willing to offer you, or work with a mortgage broker who can explore those options for you.