Botswana is the best country for a safari IMO- because there are no fences between reserves, you can still see really big herds e.g. of elephant, buffalo etc. Also, a lot of the lodges are set in private land, so when you see lions etc, it's maybe two-three cars max, rather than the 10 or more you might get in busier game reserves. Because of this, I also think it's less dangerous for the animals. The camps and the people are lovely as well (not that the people aren't lovely elsewhere). I do accept it's not a budget option, and obviously no beaches.
Areas of Kenya are dangerous- not the very touristy areas, but other parts. I don't think this is the big risk.
The biggest risk for young children is probably malaria. I believe it's not advisable to give young children anti-malarials. The NHS guidance is under 12, but I believe other countries will give them to children older than 5 or 8, but not to pre-school children. This means taking a young child to a malaria area is a real risk. Personally, I'd go to Namibia or a malaria free area of South Africa instead. Malaria is a real risk for younger children and even anti-malerials don't give full protection.
Often, there is a minimum age for game drives, usually 8 or 12. Some lodges may offer private game drives for younger children, but they'll usually refuse to stop near big cats, elephants and other dangerous animals- which defeats the point of safari for me. On safari, children will need to sit still and be quiet for long periods, as well as coping with bumpy driving at long speeds. They may also have to go for long periods without toilet stops, as these may not be safe during the game drive. At some lodges, under 12s or under 8s must share a room with an adult.
IMO, safari holidays don't really suit pre-school children, and if you do decide to go, then book very carefully to make sure that the lodge you go to can accommodate your child in all the activities you want to do.