wrap the html output in the conditional like this:
<?php $image = wp_get_attachment_image_src(get_field('post_image1'), 'thumbnail');
if( $image !=false ) { ?>
<img src="https://wordpress.stackexchange.com/questions/60013/<?php echo $image[0]; ?>" alt="<?php get_the_title(get_field('post_image1')) ?>" />
<?php }
$image = wp_get_attachment_image_src(get_field('post_image2'), 'thumbnail');
if( $image !=false ) { ?>
<img src="https://wordpress.stackexchange.com/questions/60013/<?php echo $image[0]; ?>" alt="<?php get_the_title(get_field('post_image2')) ?>" />
<?php } ?>