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.