Line after Require:
To add a horizontal line after the Require you can use \hrule:

or a rule:

\documentclass{article}
\usepackage{algorithm}
\usepackage{algorithmic}
\begin{document}
\begin{algorithm}
\caption{A good name}
\begin{algorithmic}
\REQUIRE something useful \vspace{4pt}\hrule\vspace{5pt}
\STATE $i \leftarrow 1$
\end{algorithmic}
\end{algorithm}
\begin{algorithm}
\caption{A good name}
\begin{algorithmic}
\REQUIRE something useful \vspace{4pt}\par\rule{\linewidth}{0.5pt}\vspace{5pt}
\STATE $i \leftarrow 1$
\end{algorithmic}
\end{algorithm}
\end{document}
Alternative Solution:
A better alternative might be to use \rule and incorporate the solution form Adjust spacing of algorithm float caption, which uses \DeclareCaptionFormat from the caption package:

\documentclass{article}
\usepackage{caption}
\usepackage{algorithm}
\usepackage{algorithmic}
\DeclareCaptionFormat{algorithm}{\vspace{0ex}{%
\parbox[c][1.75em][c]{\textwidth}{#1#2#3}}\par\rule{\linewidth}{0.5pt}}
\captionsetup[algorithm]{format=algorithm}
\begin{document}
\begin{algorithm}
\caption{A good name}
\begin{algorithmic}
\REQUIRE something usefull \vspace{4pt}\par\rule{\linewidth}{0.5pt}\vspace{5pt}
\STATE $i \leftarrow 1$
\end{algorithmic}
\end{algorithm}
\end{document}