The height of the posterbox header is given by the boxheaderheight key. If no posterbox header is required, set the boxheaderheight to 0em as one of the keys passed to the posterbox environment.
More generally, all the keys available in the poster environment are available to the posterbox environment. This permits extensive customization of individual posterboxes. This is illustrated by changing the posterbox header colour (headerColorOne) from blue to green and the font color for the header from white to black for the fourth posterbox.
The MWE:
\documentclass[landscape,a0paper]{baposter}
\begin{document}
\begin{poster}{
grid=false,
columns=2,
colspacing=4.2mm,
headerheight=0.10\textheight,
background=none,
eyecatcher=false,
%posterbox options
headerborder=closed,
borderColor=red,
headershape=rectangle,
headershade=plain,
headerColorOne=blue,
textborder=rectangle,
boxshade=plain,
boxColorOne=white,
headerFontColor=white,
headerfont=\color{white}\large\bfseries\sffamily,
textfont=\normalsize\sffamily,
linewidth=1pt
}
{}
{Title}
{\textsc{Author}}
{}
\begin{posterbox}[name=box1,column=0,column=0,row=0,span=1]{Title}
This box has a header. The height of the box header is given by the key \texttt{boxheaderheight}.
The default \texttt{boxheaderheight} is 2em.
\end{posterbox}
\begin{posterbox}[name=box2,column=0,below=box1,span=1,boxheaderheight=0em]{}
This box has no header. Set the height of the box header with the key \texttt{boxheaderheight=height},
where \texttt{height} is some height measure like 4em. Add the key to the options passed
to the \texttt{posterbox} environment.
\end{posterbox}
\begin{posterbox}[name=box3,column=1,row=0,span=1,boxheaderheight=0em]{}
Another box with no header.
\end{posterbox}
\begin{posterbox}[name=box4,column=1,below=box3,span=1,headerColorOne=green,
headerfont=\color{black}\large\bfseries\sffamily,bottomaligned=box2]{Title}
This box has a header. The default \texttt{headerColor1}, given in the keys to the \texttt{poster}
environment is changed from \texttt{blue} to \texttt{green}. The bottom of the box is aligned
with the bottom of \texttt{box2}.
\end{posterbox}
\end{poster}
\end{document}
The result:

\documentclassand ending with\end{document}. – Bobyandbob Jul 11 '18 at 09:21baposterboxes always declare a title part. You could leave it empty and use the background color of text part, but the space is always there. If you need nontitled boxes and you are starting with your poster a good alternative could betcbposter(fromtcolorboxpackage), it uses a boxes placement system similar tobaposterand you can use nontitled boxes. Some examples: https://tex.stackexchange.com/a/378438/1952, https://tex.stackexchange.com/a/387491/1952 and a tutorial. – Ignasi Jul 11 '18 at 09:31tcbposteryou'll have to manually select font size. It doesn't scale automatically likebaposterdoes. – Ignasi Jul 11 '18 at 09:33