The biggest cause of fatality on Kilimanjaro is altitude sickness. Therefore, the safest route will be the one which allows for the best acclimatization.
The Northern Circuit and Grand Traverse are the longest routes on Kilimanjaro. They also offer plenty of opportunities to climb high and sleep low. This is the best way to get your body used to the high altitude and reduce the severity of altitude sickness. Therefore, they have the highest summit success rates.
However, on any route, it is most important that you don’t push beyond your limits and follow an experienced guide who will assist with descent if required. If you do so, all the popular routes may be considered more or less equally safe or risky.
Besides acclimatization, the routes that avoid the dangerous Western Breach are definitely safer than the routes that don’t. To go even further, some routes also avoid the Barranco Wall. Therefore, the Northern Circuit, the Grand Traverse and the Rongai route could be deemed safest.