You need to check, if the field is empty or not before printing link using the get_the_author_meta function.
<?php if(!empty(get_the_author_meta('twitter'))) { ?>
<a href="https://wordpress.stackexchange.com/questions/75981/<?php the_author_meta("twitter'); ?>" title="Twitter" target="_blank" id="twitter"><img src="/images/twitter.png" alt="Twitter" /></a>
<?php } ?>
or, try
<?php if(!empty(get_user_meta(get_the_author_meta('ID'),'twitter'))) { ?>
<a href="https://wordpress.stackexchange.com/questions/75981/<?php the_author_meta("twitter'); ?>" title="Twitter" target="_blank" id="twitter"><img src="/images/twitter.png" alt="Twitter" /></a>
<?php } ?>
but, for some reasons, followed code fixed it
<?php if(strlen(get_the_author_meta('twitter')) >5) { ?>
<a href="https://wordpress.stackexchange.com/questions/75981/<?php the_author_meta("twitter'); ?>" title="Twitter" target="_blank" id="twitter"><img src="/images/twitter.png" alt="Twitter" /></a>
<?php } ?>