Increase width of columns

Set a length explicitly or use tabularx:

\documentclass[12pt,a4paper]{article}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{parskip}
\usepackage{graphicx}
\usepackage[top=0.65 in,bottom=0.65 in,left=0.65 in,right=0.65in]{geometry}
\renewcommand{\baselinestretch}{1.20}

\usepackage{array}
\usepackage{tabularx}
\newcolumntype{C}[1]{>{\centering\arraybackslash}p{#1}}
\newcolumntype{Y}{>{\centering\arraybackslash}X}

\usepackage{booktabs}

\begin{document}

Set a length explicitly:

\begin{center}
    \begin{tabular}[t]{|c|*{10}{C{.4in}|}}
        \hline
        Question                & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10            \\\hline
        Answer                  &   &   &   &   &   &   &   &   &   &              \\\hline
        \end{tabular}
\end{center}

Or use \texttt{tabularx}:

\begin{center}
    \begin{tabularx}{\textwidth}{|c|*{10}{Y|}}
        \hline
        Question                & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10            \\\hline
        Answer                  &   &   &   &   &   &   &   &   &   &              \\\hline
    \end{tabularx}
\end{center}

This is just to show the canonical way to typeset tables with \texttt{booktabs}, as suggested by Peter Grill:

\begin{center}
    \begin{tabularx}{\textwidth}{c*{10}{Y}}
        \toprule
        Question                & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10            \\
        \midrule
        Answer                  &   &   &   &   &   &   &   &   &   &              \\
        \bottomrule
    \end{tabularx}
\end{center}
\end{document}

Edit: one way to set the column height is to use \setcellgapes and \makegapedcells by makecell, for other methods see here: Column and row padding in tables.

\documentclass[12pt,a4paper]{article}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{parskip}
\usepackage{graphicx}
\usepackage[top=0.65 in,bottom=0.65 in,left=0.65 in,right=0.65in]{geometry}
\renewcommand{\baselinestretch}{1.20}

\usepackage{array}
\usepackage{tabularx}
\newcolumntype{C}[1]{>{\centering\arraybackslash}p{#1}}
\newcolumntype{Y}{>{\centering\arraybackslash}X}

\usepackage{booktabs}
\usepackage{makecell}
\setcellgapes{.1in}

\begin{document}

Set a length explicitly:

\begin{center}
\makegapedcells
    \begin{tabular}[t]{|c|*{10}{C{.4in}|}}
        \hline
        Question                & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10            \\\hline
        Answer                  &   &   &   &   &   &   &   &   &   &              \\\hline
        \end{tabular}
\end{center}

Or use \texttt{tabularx}:

\begin{center}
\makegapedcells
    \begin{tabularx}{\textwidth}{|c|*{10}{Y|}}
        \hline
        Question                & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10            \\\hline
        Answer                  &   &   &   &   &   &   &   &   &   &              \\\hline
    \end{tabularx}
\end{center}

This is just to show the canonical way to typeset tables with \texttt{booktabs}, as suggested by Peter Grill:

\begin{center}
    \makegapedcells
    \begin{tabularx}{\textwidth}{c*{10}{Y}}
        \toprule
        Question                & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10            \\
        \midrule
        Answer                  &   &   &   &   &   &   &   &   &   &              \\
        \bottomrule
    \end{tabularx}
\end{center}
\end{document}

You can make the columns wider by increasing \tabcolsep. Here is a before and after version of your table with \setlength{\tabcolsep}{10pt}:

Also, I would recommend you look at the booktabs package and the associated documentation. Here is an minimal example as your table doesn't really have any data:

Notes:

• I removed the {center} environment as one should be using \centering instead. See Should I use center or centering for figures and tables?.

  As CarLaTeX mentioned, with the MWE given in the question, it is ok to use the {center} environment (as it does not use the table environment). However, most will be using the table around tabular, so thought it should be pointed out.

Code:

\documentclass{article} 

\renewcommand{\baselinestretch}{1.20}

\begin{document}
\begin{tabular}[t]{|c|c|c|c|c|c|c|c|c|c|c|} 
\hline
 Question & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\\hline 
Answer & & & & & & & & & & \\\hline 
\end{tabular}  

\begingroup
\setlength{\tabcolsep}{10pt}%
\begin{tabular}[t]{|c|c|c|c|c|c|c|c|c|c|c|} 
\hline
 Question & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\\hline 
Answer & & & & & & & & & & \\\hline 
\end{tabular}  
\endgroup
\end{document}

Code: booktabs

\documentclass{article} 
\usepackage{booktabs}

\begin{document}
\begin{tabular}[t]{lcccccccccc} 
\toprule
Question & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\
%\cmidrule(lr){1-11}% <-- Not very useful here.
Answer & & & & & & & & & & \\
\bottomrule 
\end{tabular}  
\end{document}