If your garage door hinges are broken, the best thing is to replace them. While you may be able to rig up something to fix them, the safest bet is to install new ones.
There are a couple of things you need to do before you can remove the broken hinge. Lock the garage door in an open position to prevent it from falling down and injuring you. You may be able to get the hinges off while the door is down, but the springs could open the door up, resulting in injury. Also make sure that you turn off and disconnect your automatic garage door opener and any sensors that you have.
Removing and Installing Garage Door Hinges
To remove the garage door hinge, you must first loosen the nuts from the hinge bolt. Then you can take a hammer and tap the hinge bolt, releasing the hinge. Leave the nut on the end of the unscrewed bolt to protect it while tapping it out.
To install the new garage door hinges, tap the hinge bolts through the bolt holes, preferably from the outside. Once tapped in, tighten the nut.