SI inserts line break before unit

I'm not feeling at all comfortable with the way you are putting a whole display math and even normal text in a \bm argument. Imho you should at least try to detect math mode.

\documentclass{article}
\usepackage{xcolor}
\usepackage{bm}
\usepackage{siunitx}

%% https://tex.stackexchange.com/questions/64547/getting-bm-to-pass-thru-siunitx-macros
\sisetup{detect-weight, detect-display-math}
\sisetup{detect-inline-weight=math}

\newcommand\SetInBoldFont[2]{%    
    \begingroup
    \color{#1}%
    \ifmmode
     \bm{{#2}}% either additonal braces here or at use around fragile commands.
    \else
     \bfseries\mathversion{bold}#2
    \fi
    \endgroup
}

\def\SIText{\SI{1}{\kilo\meter}}

\begin{document}
In math mode things work fine:
\SetInBoldFont{red}{%
\[
   \SIText \text{ where $\si{\kilo\meter}$ is kilometers}
\]
}
Inline math also works: $\SetInBoldFont{red}{\SIText} $

And Outside of math mode: \SetInBoldFont{blue}{\SIText}

\end{document}

This is similar to Ulrike’s solution, but more robust, see the final part.

\documentclass{article}
\usepackage{amsmath}
\usepackage{bm}
\usepackage{xcolor}
\usepackage{siunitx}

%% https://tex.stackexchange.com/questions/64547/getting-bm-to-pass-thru-siunitx-macros
\sisetup{detect-weight, detect-display-math}
\sisetup{detect-inline-weight=math}

\newcommand\SetInBoldFont[2]{%
  \begingroup
    \ifmmode
      \textcolor{#1}{\bm{{#2}}}%
    \else
    \renewcommand{\seriesdefault}{\bfdefault}%
    \bfseries\boldmath\textcolor{#1}{#2}%
    \fi
  \endgroup
}

\def\SIText{\SI{1}{\kilo\meter}}

\begin{document}
In math mode things work fine:
\SetInBoldFont{red}{%
\[
    \SIText \text{ where $\si{\kilo\meter}$ is kilometers}
\]
}
Inline math also works: $\SetInBoldFont{red}{\SIText}$

Outside of math mode things don't break:
\SetInBoldFont{blue}{\SIText}

\textit{\SetInBoldFont{blue}{This is in italics \SIText}}

\SetInBoldFont{blue}{\textit{This is in italics \SIText}}

\SetInBoldFont{blue}{\normalfont\itshape This is in italics \SIText}

\end{document}

By the way, \let\text\textbf is wrong under many respects. The main one is the usual business described in the documentation of letltxmacro. Here, however, there's a much more important aspect.

How does amstext (that’s loaded by amsmath and is responsible for defining \text) makes \textbf to change size when in subscripts or superscripts? Well, if \textbf{foo} appears in math mode, LaTeX executes \nfss@text{\textbf{foo}} and \nfss@text is \mbox when amstext is not loaded. But amstext does \let\nfss@text\text: can you see the infinite loop? ;-)

Another "idiot solution", which simply uses text mode in all cases. This approach also simplifies the \sisetup stuff.

\documentclass{article}
\usepackage{xcolor,siunitx}

\newcommand\SetInBoldFont[2]{%
     \begingroup
     \sisetup{detect-weight}%
     \textbf{\color{#1}#2}%
     \endgroup}
\def\SIText{\SI{1}{\kilo\meter}}

\begin{document}
In display math things work fine:
\[\SetInBoldFont{red}{\SIText}\]

In inline math things work too: 
$\SetInBoldFont{purple}{\SIText}$

In text mode things are also fine:
\SetInBoldFont{blue}{\SIText}
\end{document}