In order to improve your internet connection in the basement, you can try a number of things:
-run a dedicated cat5 line from your router into the basement (this would be the best solution, as a hardwired connection is far more reliable than a wifi connecction; also improves the latency - which is very important in video games)
-us a range extender in order to improve the wifi reception in the basement (there is a ton of them available everywhere; they are easy to set up; not as good as a hard line connection though)
-move the current wifi router somwhere closer to the basement location (maybe move it --if possible-- from the second floor to the first floor)
-use a powerline adapter - which uses your existing electrical outlets as a means of transferring data (This involves two adapters, one by the video console, and one by the router. It's the next best option after running a dedicated cat5 line, as it creates a hard wired connection for your network. Keep in mind that in order for a powerline adapter to work you can't have more than 1 elecrical panel, and the adapters can't be connected to a power strip)
Either way remember that a hard wired connection is always the way to go. Latency is super important in video gaming, especially in shooter games, and wifi is your worst nightmare when it comes to the latency. Also certain home appliances such as washers, driers, microwaves will create a magnetic field of its own, and thus cause
interference to your wifi network.