I created an Image Field for the Basic Page Content Type.
I want to control the output of this image to use as a banner with a caption on top. Is there a function to render out a field specific to the page,
such as <?php print render($page['my_image_field']); ?>.
I wan't to style this separately from the body content.
You can do this within a tpl file specific to your content type. In this case it sounds like it'd be node--page.tpl.php
All the drupal 7 template naming suggestions are available at - https://drupal.org/node/1089656
Just make sure you have a regular node.tpl.php file in your theme before creating the new one.
From there this thread tells you how to go about printing your field elsewhere (specific to drupal 7)- https://drupal.org/node/1059636
Specific to your question:
<?php
hide($content['field_my_image_field']);
print render($content);
?>
then wherever you want your image to appear at use
<?php
print render($content['field_my_image_field']);
?>
