The main difference is that "base" is the literal word and "basis" the figurative word. So "base" means the foundation/the bottom part. If you want to build something on the ground and you want it to be firm and not move, you must make sure that the base is solid. If you are talking about an argument/a discussion/a philosophy, you would use the word "basis" for the starting point.

Simple answer in the context you are talking about is that "base" is the main verb, "to base". It is easier to use these two words in a sentence to show the difference: Question: "What is the basis of you opinion?" Answer: "I base my opinion in fact"