like this?

\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{empheq,
amssymb}
\usepackage[dvipsname]{xcolor}
\newcommand*\widefbox[1]{\colorbox{teal!30}{\qquad#1\qquad}}
\begin{document}
\begin{empheq}[box=\widefbox]{equation}
\begin{multlined}
\sum_{k=-\infty}^{\infty}
\frac{(-1)^k(\alpha+k+\frac12) \sin \sqrt{b^2+\pi^2(\alpha+k+\frac12)^2}}
{(\beta^2+(2k+1)^2)\sqrt{b^2+\pi^2(\alpha+k+\frac12)^2}} \\
= \frac{\pi(\alpha+\frac12 i \beta)
\sin\sqrt{b^2+\pi^2(\alpha+\frac{i}{2}\beta)^2}}
{4i\beta\cosh(\frac{\pi}{2}\beta)
\sqrt{b^2+\pi^2(\alpha+\frac{i}{2}\beta)^2}} \\
- \frac{\pi(\alpha-\frac12 i \beta)
\sin\sqrt{b^2+\pi^2(\alpha-\frac{i}{2}\beta)^2}}
{4i\beta\cosh(\frac{\pi}{2}\beta)\sqrt{b^2+\pi^2(\alpha-\frac{i}{2}\beta)^2}}.
\end{multlined}
\end{empheq}
\end{document}
or you like to have frame and colored text?

\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{empheq,
amssymb}
\usepackage[dvipsname]{xcolor}
\newcommand*\widefbox[1]{\fbox{\qquad#1\qquad}}
\begin{document}
\begin{empheq}[box=\widefbox]{equation}
\color{red}
\begin{multlined}
\sum_{k=-\infty}^{\infty}
\frac{(-1)^k(\alpha+k+\frac12) \sin \sqrt{b^2+\pi^2(\alpha+k+\frac12)^2}}
{(\beta^2+(2k+1)^2)\sqrt{b^2+\pi^2(\alpha+k+\frac12)^2}} \\
= \frac{\pi(\alpha+\frac12 i \beta)
\sin\sqrt{b^2+\pi^2(\alpha+\frac{i}{2}\beta)^2}}
{4i\beta\cosh(\frac{\pi}{2}\beta)
\sqrt{b^2+\pi^2(\alpha+\frac{i}{2}\beta)^2}} \\
- \frac{\pi(\alpha-\frac12 i \beta)
\sin\sqrt{b^2+\pi^2(\alpha-\frac{i}{2}\beta)^2}}
{4i\beta\cosh(\frac{\pi}{2}\beta)\sqrt{b^2+\pi^2(\alpha-\frac{i}{2}\beta)^2}}.
\end{multlined}
\end{empheq}
\end{document}
note: since the equation is to wide that its number can be placed in line with equation, i broke it into three lines using multlined environment from mathtools (which is loaded by the package empheq). if yout liked this format of equation, you can insert your equation as you have with split environment.
if you like to change box line color and lines thickness than use fcolorbox (from the package cxolor). for example:
\newcommand*\widefbox[1]%
{
\setlength{\fboxrule}{2pt}% if you like to change box lines thicnes
\fcolorbox{blue}{white}{\qquad#1\qquad}%
}
and you will obtain:

(well, to my taste, this is to fancy :-( ).