No, you don’t have to wait until the garage falls down to file a claim, but whether home insurance covers the repairs depends on your policy, as well as what caused the damage and when.
If the damage is a result of a peril that’s covered by the policy, such as fire or a hailstorm, then home insurance should theoretically cover the cost of repairs. In that case you should file a claim as soon as possible. The claim may be denied if you wait too long.
But if the garage is unstable because of lack of maintenance over the years or sloppy construction, then it’s your responsibility to pay for fixing it. Home insurance generally does not cover damage that stems from maintenance issues, such as termite infestation or water damage from a slow leak, because insurers figure you could have prevented the problem.
Most perils that aren’t your fault and are not tied to maintenance issues are covered by standard home insurance, but there are exceptions. Earthquakes and floods, for instance, are not covered by home insurance. You must buy separate policies for protection from those disasters. Read your home insurance policy and talk to your insurance agent to understand what’s covered and what’s excluded.
Regardless of whether insurance covers repairs to the garage, do whatever is necessary to ensure the safety of your family and anyone else who visits your home. If the garage is as unstable as you say, then do whatever is needed to stabilize the structure and prevent it from collapsing on someone.
For more, see home insurance basics.