How to add a dot after the section number?

You can do this simply with the titlesec package:

\usepackage{titlesec}
\titlelabel{\thetitle.\quad}

It depends on what you wish precisely. If you want that in all places the section number appears as "1.1.", also in cross references, the question is settled quite easily:

\renewcommand{\thechapter}{\arabic{chapter}.}
\renewcommand{\thesection}{\thechapter\arabic{section}.}

(in this case you would also change chapter numbers, for uniformity).

If you want only the number in the section title to be followed by a period, then you can follow Alan's good suggestion or delve into the internals (see the FAQ entry for more information):

\makeatletter
\renewcommand{\@seccntformat}[1]{\csname the#1\endcsname.\quad}
\makeatother

This will change the format of all the sectional units also below \section. You might prefer the titlesec way to get more control on the appearance of the section headings.

If you use Komascript you may use the option numbers=endperiod.

Example:

\documentclass[11pt,english,numbers=endperiod]{scrartcl}
\usepackage{babel}
\usepackage{blindtext}

\begin{document}
\tableofcontents
\blinddocument
\end{document}