Quebec is an extremely safe city, I wouldn't be afraid of leaving my car anywhere, except for maybe a couple of shady areas that tourists don't ever get to. Take the usual safety precautions - don't leave any visible objects in your car and lock your doors, but other than that, there's no need to worry, any parking you find should be absolutely safe.

Other options - take the bus. If you're flying in, you'll need to take a cab to and from the airport and it's expensive (it's pretty annoying that we still don't have any kind of shuttle!), but other than that, the buses work great and will take you anywhere you need to go, unless you're looking at day trips outside the city and its suburbs. The RTC offers several types of passes that you could look into.

Otherwise, stay outside the center. It takes about 10-15 minutes by bus to get from the hotels on Boulevard Laurier to Old Quebec. The location isn't as interesting, but depending on how long you're staying, the savings might be worth it, since both your hotel room and parking would be way less expensive than in the center (in fact, I'm pretty sure parking is free at most of these hotels).

Whatever you do, try and avoid cabs as much as possible, as they get expensive quickly when you factor in tips and the like, especially when the buses work just as well! Enjoy your trip!