Battery problem?

1995 Mitzubishi Mirage, 100,000 miles, 1.5 L engine. This is my son's car. He has had to jump start the car every day lately. Even if he lets it sit for 45 minutes after driving it, it won't start without a jump. I bought a new battery and it still will not start.

I thought that it might be the alternator draining the battery, but shouldn't it start at least once when the battery is brand new? Nothing works, the lights don't go on, so I thought maybe it was the battery cables, but why will it start with jumper cables?