Stop worrying!
Malaria is non-existent in Bangkok. Dengue is easily prevented with a once a day application (daytime) of insect repellent. Rabies, well, just teach your kids not to approach stray dogs. Same as in any country. I've never heard of anyone here getting rabies. If you do get bitten, straight to the hospital for a wound clean out and a rabies shot. Hospitals are like 5 star hotels.
Bangkok is incredibly safe. I feel much safer here than in London as far as robbery, theft, burglary goes. I have never heard of an expat home being broken into, especially if you live in a condo. I've never known anyone be mugged, a couple of handbag thefts in bars, but that's it.
Btw, it might be poor in general, but there are a lot of very rich Thais living in Bangkok, especially in areas such as Sukhumvit and Sathorn, as well as Bang Na and Nichada. Lamborghinis and Porsches are not that rare, Mercedes and BMWs are ubiquitous. The malls are full of people buying Chanel, Hermes and Prada. It's not exactly living somewhere like Darfur. Trust me, you won't be a target. Not unless you are richer than Madonna.
Just live your life with the same safety precautions that you would in the uk and you will be fine.
I hate gated complexes and the burbs, don't see the point of them and find them extremely dull and isolating, but each to their own. If you do live outside the city centre, a car is a must. Otherwise, you will have to risk taxis, and they might be cheap, but they don't have car seats (they don't in the uk either, for the most part). Bring your own car seat if you can, they are expensive here in comparison.
It's not exactly a hardship posting - many expats here have no intention of ever returning home. You just have to be open minded to different food, cultural norms, etc.
Do you have somewhere to live sorted? Schools? Happy to help with advice, feel free to PM me.