Diagnosis
This is almost always a connection issue between the handle and the charging base, not a dead battery. A tiny bit of water or toothpaste residue can break the connection.
How to Fix It
First, unplug the charging base from the wall for a full minute. While it's unplugged, grab a dry cotton swab and some isopropyl alcohol. Dab the swab and thoroughly clean the two metal charging pins on the bottom of the toothbrush handle and the corresponding contacts on the base. Let everything dry completely for five minutes. Then, plug the base back in and seat the handle firmly. The light should come on.
If That Didn't Work
Try charging it with a different USB power adapter and cable, if you have one. The one that came with it can sometimes fail. Use a phone charger block that you know works.
Heads Up
Don't store the handle directly on a wet sink. The moisture under it creates a perfect path for corrosion on those pins. Always dry the bottom of the handle and the base before charging.