Storing a lawn mower in the basement with gas in its tank can be potentially dangerous. While a mower's fuel tank should limit the amount of fumes and fuel that leaks when the mower is stored, even a small amount of gasoline liquid can produce enough fumes to make an enclosed space like a basement uncomfortable to endure. Additionally, gas fumes can potentially be hazardous to your health and, in sufficient quantity, can be explosive.
If you intend to store your lawn mower in your basement over the winter, you may be able to sufficiently empty the gas tank so as to limit the amount of odor you bring into your basement. Empty the gas tank as best you can manually, using a gas siphon. Following that, run the mower until it runs out of fuel, to burn off any gasoline that may be in the carburetor or fuel lines. Leave the mower outside with the gas cap off for an afternoon to allow the tank to fully air out.
Add Very Little Gas
If you must experiment with storing your lawn mower in the basement, you will want to limit the amount of gasoline you put in the mower's tank for each usage. Start by adding a very small amount, less than half of a cup. For future uses, figure out how much gasoline it takes to mow your lawn, and add slightly less than that amount. Every time you mow your lawn, before storing the mower in your basement, run it until it stalls out from lack of gas, and then leave it outside with the gas cap off for a few hours. If, once you have the mower in your basement, you smell a gasoline odor, remove the mower immediately and find different accommodations for it.
If your basement has windows that can be kept open for ventilation, use them. While gasoline from a lawn mower is unlikely to pose the risk of an explosion, it can stink up your basement pretty quickly. Keep your basement as well ventilated as possible, and if there are any gas odors detectable in your house, remove the lawn mower immediately.
A Few Words of Caution
Never run your lawn mower in an enclosed space such as a basement. In a very short amount of time, it can produce sufficient carbon monoxide to cause you to lose consciousness. Do not store the container of gasoline for your mower in the basement. Gasoline expands and contracts according to the temperature, and can pop off the built-in vents in gas cans and leak fumes into your basement.