The solution to your problem depends on the terrain and the location of the property line.
Putting up a dam along the edge of your driveway will not solve the problem unless you have a place to divert the water. If the water doesn't have a place to go, it will eventually soak in under your new driveway. That is not a good place to dispose of the water.
If you can't direct the water along the driveway to the point where it will disappear downhill, you will have to find a good way to get it safely across your driveway.
If you have enough room, you should be able to put in a swale (shallow ditch with gentle slopes, that you can negotioate with a lawn mower) that will carry the water away.
If you don't have some slope near the driveway, your solution might require an underground pipe to a low place.
Someone with a surveying instrument, or perhaps a rotating laser level, could figure out what you need to do to make the water go away.