The importance of basis constant in Banach spaces

Let $X$ be a Banach space an let $(e_n)_{n=1}^{\infty}$ be a Schauder basis of $X$. If we denote the sequence of the natural projections associated with $(e_n)_{n=1}^{\infty}$ by $(S_n)_{n=1}^{\infty}$ then by Uniform Boundedness principle we can observe that $$\sup_n||S_n||< \infty.$$

And the number $K=\sup_n ||S_n||$ is called basis constant.

At this point, I have a question. What is the importance of the basis constant $K$? What happens when $K=1$?

A basis with basis constant 1 is called monotone. You can always find an equivalent norm such that the basis constant is 1. It is very simple: Simply define the new norm as $|||x|||=\sup_n \|S_nx\|$.

A more interesting question is whether this new norm satisfies any "desirable" properties, such as uniform convexity. There is some literature on questions of this type.

Your first question is too general. As Michael pointed out, there is always an equivalent norm under which the basis has basis constant one, and sometimes you can choose the equivalent norm to have good geometric properties. As it is nice to have approximation of the identity operators by finite rank projections of norm one, if one does not care much about keeping the original norm it is better to change the norm to make the basis constant one.

For a theorem in which the basis constant plays a role, the Gurarii-Gurarii-James theorem relates the basis constant to the existence of upper and lower $\ell_p$ estimates of linear combinations of basis vectors. See e.g. M. Fabian's book "Functional Analysis and Infinite-Dimensional Geometry".

In some classical spaces, bases with constant one have very nice properties. For example, in Hilbert spaces only orthogonal bases have constant one. In $L_p(0,1)$, $1<p<\infty$, a basis for the entire space that has constant one is a disguised martingale difference sequence and hence is unconditional. See

Dor, L. E.; Odell, E. Monotone bases in Lp. Pacific J. Math. 60 (1975), no. 2, 51–61.

I know one situation where the basis constant plays a role. This is the Paley-Wiener theorem on the perturbation of bases (nota bene, not the more famous one on the characterisation of the Fourier transform of $L^2$ functions). It is a quantitative version of the intuitively obvious result that the isomorphic character of a Banach space is stable under small perturbations. As such it played an important role in the initial explosion of the isomorphic theory of Banach spaces in the 70's (structure of $\ell^p$-spaces and their subspaces, resp. quotient spaces).