Using a conditional tag for the home page banner

There are multiple issues that I think you want to adjust:

1. You open an extra PHP tag before the second if statement. That shouldn’t be there.
2. Make sure that is_home() is what you want. If you have a static front page (set in Settings > Reading), then you actually want is_front_page(). Think of is_home() as is_blog() (to be clear, is_blog() does not exist).
3. If you want to give the_post_thumbnail() a width and height, it needs to be in an array. Your other option is to define an image size in functions.php and then call that there.
4. I think your HTML can be cleaner by just always wrapping the banner in the #banner div.
5. Just to be sure that you always have an image, I would check to make sure that you have a post thumbnail set and make sure you have a fallback.

With all those suggestions wrapped into code, here’s what I suggest:

In your functions.php add:

add_image_size( 'banner', 863, 328, true );

And wherever your banner snippet is:

<!-- Let's always wrap our banner in the banner div for easy styling -->
<div id="banner">
<?php

// Check to see if we're on a non-home-page and if the non-home-page has a featured image. use is_front_page() if that's desired
if( !is_home() && has_post_thumbnail() ) {

    get_the_post_thumbnail( 'banner' );

// if we're on the home page OR don't have a featured image but DO have a dynamic sidebar
} elseif ( function_exists('dynamic_sidebar') && is_dynamic_sidebar('banner-widget') ) {

    dynamic_sidebar('banner-widget');

// no featured image or on homepage with no sidebar, spit out a static image
} else {
    echo '<img class="banner" src="' . get_template_directory_uri() . '/img/banner.jpg" alt="Banner 1">';
}
?>

UPDATE:

That final echo statement was previously a complete mess. Sorry! I cleaned it up and I think it should work now. I also replated the bloginfo() function (which is now deprecated) with the new, preferred equivalent function.