Page split between longtable caption and table

Assuming you want to have at least 3 rows of the longtable to show up before considering a page break to be unacceptable, you could load the needspace package and issue the directive

\Needspace{5\baselineskip}

immediately before \begin{longtable}. That way, unless there are at least 5 lines of text still free at the bottom of the page, the longtable will start at the top of the next page.

Why 5\baselineskip? Because the longtable's caption and the blank line between the caption and the start of the body of the text take up 2 more lines.

You can simply use \\* instead of \\ after the \caption. From the manual:

\\*: The same as \\ but disallows a page break after the row.

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longtable has some macros to control headings (cf. manual section 3). Depending on your use-case, have a look at \endhead or \endfirsthead. This code works for me:

\documentclass{report}
\usepackage{lipsum}
\usepackage{longtable}

\begin{document}
\lipsum[1-4]

another paragraph

another paragraph

another paragraph

another paragraph

anoteher paragraph

\begin{longtable}{lr}
  \caption{\lipsum[1][1-3]}\\
  \hline
  \endfirsthead
  A&B\\
  c&D\\
  E&F
  \end{longtable}
\end{document}