Intuitively, the volatility of basis is the volatility of your hedging portfolio( long spot, short future).  So when you hedge the spot, you reduce the risk. The volatility is risk. So the volatility of basis is much lower than either the spot or the future. Otherwise there is no meaning to do the hedge.

A more formal explanation is that future price equals spot price plus something we call the carrying cost. F=S+C.  For example, if we just consider the financial cost, the future price is F=Sexp(rT), basis is S(exp(rT)-1), we know that Var(S(exp(rT)-1))=(exp(rT)-1)^2*Var(S), rT is usually very small, so exp(rT) is close to 1 and exp(rT)-1 is a very small number. So Var(basis)=Var(Spot)*a very small number, at least much smaller than 1. So the volatility of basis is much lower than the spot's or the future's (futures basically have a higher volatility than spot since Var(F)=Var(S)*exp(2rT)>Var(S))

I hope I have explained it clearly.