I will be traveling to Basel, Switzerland in June and will be staying there for approximately 1 week. While there, am I free to visit neighboring Germany and France freely?
Generally speaking the borders between Switzerland and its neighboring countries Liechtenstein, Austria, Germany, France and Italy are open for every one. Obviously, you need an appropriate visa, but if that is okay there is nothing that restrains you from crossing any border.
For example with a Schengen visa you can cross every border between Switzerland and its neighboring countries.
Yes. As a citizen of the United States (I presume), you are in the visa-exempt ("Annex II") category, and will receive a landing permission on arrival that is valid for 90 days throughout the Schengen area, which includes Switzerland, all its neighbors, as well as everything else in the map below (thanks Wikipedia!) that is blue or light blue.