It really varies. The kinds of places you will find lots of attractions and restaurant tend to be pretty safe. Stay on the Missouri side. University City Loop and Central West End are very popular areas. CWE is nearer some 'family' attractions like Forest Park which has the Zoo, Science Center, Art Museum, History Museum. It's also near the City Museum which is kind of an oddball museum that I think caters to kids, though it does have a small bar inside of it as well. A little further, but not far is, of course, the Arch.
If you venture west, in St Charles county, there is a series of wineries and a brewery (Augusta Brewery is GREAT for beer and food) along highway 94, which follows the river and the Katy Trail. That's the south end of St Charles. On the north end is old St Charles City, which (is also on the Katy Trail and) has an old main street with lots of neat shops and restaurants. I think there are a couple of breweries, one being Trailhead. I'm sure they have shops that sell many of the Missouri wines as well.
I'm not sure about activities for kids there, though.
Most of the places you would choose to take your family would be pretty safe. I live downtown (Lafayette Square area) and my wife and I walk up to the local brewpub all the time. The main thing is just to use common sense and be observant of your surroundings. Bad stuff can happen anywhere, at any time of day. Minimize your risk by following the above advice and following your instincts.
The previous poster mentioned some great attractions. The Magic House is also a lot of fun... we took our neice and nephew there last summer. Its kind of like a science center. Groupon has some great deals sometimes.
The City Museum is a blast for kids! It sounds boring, but the name is deceiving. They buy all kinds of 'junk' from all over the city and weld it together so kids can climb/play on it. I mean BIG stuff. Where else can a kid climb into a school bus that is on the roof of a building? You can check it out here: CITY MUSEUM- Home
Again, if you are sight seeing, just use common sense. Popular, family-friendly attractions are well-lit and well-patrolled. Remember also that there is safety in numbers. Stay where the people are and you'll be fine.
Forgot to mention an interesting place. In Chesterfield (just north on Clarkson Rd from Hwy 40/64) is a place called the Butterfly House. It may not be worth a trip all the way out to Chesterfield on its own but it's nice if you don't mind some butterflies landing on you. And the kids will probably love it. If you're sans-kids, you and the wife can visit the Inernational Tap House in Chesterfield as well. Or you can visit the longest strip mall in Chesterfield. *sigh*
Also, coming from down town, Chesterfield is conveniently on the way to the wineries down hwy 94.
I would recommend you stay away from any part of NORTH St. Louis. That would mean don't go north of the Edward Jones Dome and don't go further south than highway 40/64. As a poster above said, the Butterfly House in Chesterfield is really wonderful and Chesterfield is very safe (as compared to downtown). You must use your wits and be aware of your surroundings while downtown. DO NOT leave anything of ANY VALUE in your car....especially in plain view. If you do you may come back to shattered windows and find all your goodies gone!!!!!!!! I never go downtown without my concealed carry sidearm. Just my .02 cents worth.
I can vouch for the Butterfly House. Took my wife and twins there a couple years ago and they loved it!
I can also vouch for the City Museum. Lots of fun for kids....and those of us that are big kids at heart. I personally love the zoo section the most. It'll be a blast for the whole family. There's a TON of art....so pay attention to your surroundings if you go.
Of course, if you like Italian, it's hard to beat The Spaghetti Factory. There are three in the area, but I've only been to the one downtown and the one out off Hwy 40 somewhere. Either one are wonderful!
If you like historic towns, Old St. Charles is really hard to beat! We go back there time and time again. Great shops, great location, and awesome restaurants. Our twins are older (19)...so it might not be that great for your kids if they're much younger.
The Science Center....wow! Awesome place, too!
Can we get a little more information so it'll help make better recommendations? How old are your kids (assuming you have some)? I know you said you wanted to know what areas to avoid, but what's been said pretty much covers it. If you're aware of how the most avoided areas look in big cities, the same common sense comes in to play, here.
Like most people said, be aware of your surroundings. I travel all over the city on the job and feel safe, but I also limit my time in parts of the city from early morning to noon. I try not to leave valuables in plain sight anywhere I travel, just keeping honest people honest. Don't be afraid to be in our great city, it's a great place to visit with a lot of historical value. Jefferson Barracks in South County is a national cemetary, and Bellefontaine cemetary is where many prominent St Louisans are buried. The architecture of the mausoleams is spectacular.