It's becoming code in a lot of places to vent your crawl space. Pretty much everywhere if you live in a flood plain in order to get insurance. Whether flood plain or not, many areas are requiring them for everything new. Vented helps to prevent both mold and buildup of gasses like radon in your house. You can get vents which open and close according to the temperature (and water pressure in case of a flood) if you think that would help. People post mixed reviews as to the benefit of these products. Do your research if you go this route.
It is perfectly acceptable and I would say, the best approach to spray foam insulation (similar to "Great Stuff") in a crawl space that is properly prepared for it. It needs to be vented already. It needs to have certain things like pex pluming covered as the foam will eat it. The foam goes right up to your sub-floor adhering to the joists and down the walls a little. And it does wonders for your energy bill.
I owned a 4k square ft house in Montana in a flood plain with 3k square ft of crawl space that was vented with flood vents (automatic open and closing vents). I had the entire 3k square ft of space sprayed with 3" of foam insulation and it cost me about $2800. This was in 2008. -30 degrees Fahrenheit and the floor stays fairly warm. An added benefit is this does wonders for bug protection, completely sealing up your floor. This is the only way to go in my opinion.