Kenya and Tanzania are great safari destinations all year round. The major difference is in the seasons that influence prices.
June, July, August and September are considered high season- with July and August being peak. This is mainly because of the Great Wildebeest Migration that attracts many from far and near.
IMO the best time for a family with kids would be January to March or between 1st to 20th December when it is not too crowded and there are great wildlife sightings as well.