When a driveway slopes down into the house, you need to intercept any and all water flow and carry it away before it gets to the garage. If you don't do this, you will find the concrete pad in the driveway getting into moisture and frost problems, the door freezing to the floor, and the like.
You may have seen the grilled trenches in commercial installations and in some garages, but it really should be in all driveways that slope toward the house. Markant by ACO, from Mississauga, Ontario, makes a whole variety of trench drains for both heavy duty and light duty use. In fact they specialize in controlling water run-off problems all around the house. They come in sections; have leaf catches and removable grills. You can put them in driveways or even in landscaping areas where you need to divert water. You drain them to a ditch, a sump pump or to a storm drain system, a catch basin or whatever is appropriate for your house and your municipality.
The third photo shows the details. Note how the outflow pipe is turned down into the trench. That keeps leafs from clogging the drain as they will float above the point where the water enters the pipe. They can be installed after the fact but of course much easier to do it during the construction of the driveway.
If you are having problems of freezing of any bottom of the driveway drainage system you could use salt, or better yet non-corrosive de-icers to keep them open. If it is really a chronic problem you can use plumbing heating cables with built in thermostats to keep the trap and the line open until it gets out of the freezing area - which could mean right through an un-heated garage.