We live in the basement of my sister's home. Our closet is right up against another closet that has the water heater in it. A couple weeks ago, we noticed a smell coming into our closet. It smelled like old broccoli and was pretty strong. We turned the water heater off and then a few hours later turned it back on again. The smell went away after turning it off and didn't come back immediately after turning it back on. Then, a few days later, the smell came back again. Any thoughts??

If you shut the gas off and the smell went away and it came back after you turned it back on I would suspect a gas leak or a flame setting that didn't burn all the gas. Is there amy smoke?
In any respect if any gas appliance begins to have a odor it's time to have it checked out by a pro.
Better safe then sorry! Good luck, Tom