If mass were basic, then would it be convertible into something else, such as energy? What they do is rotate their dimensional unit matrices until they become relatively simple; then one of the dimension units becomes mass. But there are other possibilities.
Perhaps I did not explain myself well. Tthe idea or concept of mass is "basic" or "fundamental" in the sense that it requires no advanced mathematics to understand. (Most humans have at least an vague idea that objects feel heavy and resist motion.) As I am using it, basic does not mean that the mass of an object is unchangeable.
Perhaps a better word than "basic" would be "axiomatic."
Other basic or fundamental physical concepts are length, time, and (perhaps) force. Most people have an intuitive feel for these concepts; they cannot really be explained in terms of simpler concepts.
BTW, under what circumstances is mass convertible into energy? Einstein's famous mass-energy equation (the one equation everyone seems to have seen) implies not that mass and energy are interconvertible, but rather that the mass of a body is a measure of its energy. Increase the energy, you increase the mass.