The risk of a tree or its parts failing can never be completely eliminated. It can, however, be reduced to a reasonable level so that you can have peace of mind while enjoying the benefits that your trees provide.
Proper tree care is the single best course of action.
In the short term, pruning is the best tree care practice for addressing immediate tree hazards. This includes removing high-risk branches, such as those that are dead or declining. Crown reduction and crown thinning are specialized pruning techniques that may help reduce wind damage during extreme weather. Periodic pruning also helps develop stronger branch attachments that withstand wind, ice, and snow better. Tree topping is improper pruning and is not effective in reducing long-term tree risk. Other potential practices to address immediate hazards include cabling and bracing to reinforce weak tree parts.
Over the long term, mulching, irrigation (if needed), and fertilization (as prescribed by a soil test) help build healthy root systems. Proper mulching is perhaps the cheapest and best overall action anyone can do. It improves soil conditions for root development, which helps reduces the risk of root failure and thus the failure of a whole tree. Check out this great YouTube video on mulching from Mississippi State.