I'm not sure this is what you want, but for what it's worth:
The basis constant can be characterized as the smallest $K$ so that for any choice of scalars $(a_n)$ and any integers $m<n$ we have
$$
\biggl\Vert\sum_{i=1}^m a_i e_i\biggr\Vert\le K \biggl\Vert\sum_{i=1}^n a_i e_i\biggr\Vert.
$$
The value of $K$ gives a measure of how "orthogonal" the basis is in the sense that it measures how drastically the basis elements can cancel with each other. (In fact, in Hilbert space, a basis has constant $K=1$ if and only if it is orthogonal in the Hilbert space sense).
The value of $K$ is also used in the hypothesis of various theorems, specifically "perturbation theorems". For example: if $(x_n)$ is a normalized basis of $X$ with basis constant $K$ and if $(y_n)$ is a sequence in $X$ such that $\sum\Vert x_n-y_n\Vert <1/(2K)$, then $(y_n)$ is a basis of $X$ equivalent to $(x_n)$.
One can also obtain a bound on the norm of the associated coefficient functions from the norms of the $e_i$ and $K$.
Basis with basis constant $K=1$ are called monotone and are special if only for the fact that not every Banach space with a basis has one.
Also, re the above comments, monotone bases were originally called "orthogonal bases".
If $K=1$, the norm of $\sum_{i=1}^m a_i e_i$ cannot be decreased by adding a linear combination of basis elements $e_j$, $j>m$.
The value of $K$ will not in general indicate if a given basis is unconditional.
(In fact, given a Banach space $X$ with a basis, there is also a conditional basis of $X$. $X$ may be given an equivalent renorming so that this basis is monotone (as is the case with any basis).
Finding specific examples of a conditional monotone basis is not hard (e.g., in $\ell_1$: $x_1=e_1$, $x_n=e_n-e_{n-1}$, $n>1$).)
All the above can be found in Ivan Singer's Bases in Banach Spaces, Vol. 1.
