i have edited your code a bit. And i think it should work.You need to pass parameters to previous_posts_link and next_posts_link.
So try the following code
<section id="blogposts" class="content-block content-3-5 blogposts">
<div class="row">
<!-- Feature Box 1 -->
<?php
$paged = (get_query_var('paged')) ? get_query_var('paged') : 1;
$blogposts_args = array(
'post_type' => 'post',
'post_status' => 'publish',
'nopaging' => false,
'order' => 'DESC',
'orderby' => 'date',
'posts_per_page' => '12',
'paged' => $paged
)
?>
<?php $blogposts = new WP_Query( $blogposts_args ); ?>
<?php if ( $blogposts->have_posts() ) : ?>
<?php $blogposts_item_number = 0; ?>
<?php while ( $blogposts->have_posts() ) : $blogposts->the_post(); ?>
<?php
$thumb = wp_get_attachment_image_src( get_post_thumbnail_id($post->ID), 'thumbnail' );
$url = $thumb['0'];
?>
<div class="col-md-3 col-sm-6 col-xs-12">
<div class="feature-box">
<div class="column-item-category"><?php the_category(); ?></div>
<a class="feature-box-link" href="https://wordpress.stackexchange.com/questions/200313/<?php the_permalink(); ?>">
<?php the_post_thumbnail( '', array(
'class' => 'column-circle'
) ); ?>
</a>
<a class="feature-box-link" href="https://wordpress.stackexchange.com/questions/200313/<?php the_permalink(); ?>">
<h4 class="column-item-title"><?php the_title(); ?></h4>
</a>
<p class="column-item-author"><?php the_author(); ?></p>
</div>
</div>
<?php $blogposts_item_number++; ?>
<?php if( $blogposts_item_number % 4 == 0 ) echo '<div class="clearfix visible-md-block visible-lg-block"></div>'; ?>
<?php if( $blogposts_item_number % 2 == 0 ) echo '<div class="clearfix visible-sm-block"></div>'; ?>
<?php if( $blogposts_item_number % 1 == 0 ) echo '<div class="clearfix visible-xs-block"></div>'; ?>
<?php endwhile; ?>
<?php wp_reset_postdata(); ?>
<div class="post-nav-links">
<div class="prev-post"><?php previous_posts_link('Newer', $blogposts->max_num_pages ); ?></div>
<div class="next-post"><?php next_posts_link('Older', $blogposts->max_num_pages ); ?></div>
</div>
<?php else : ?>
<p><?php _e( 'Sorry, no posts matched your criteria.' ); ?></p>
<?php endif; ?>
<!-- Feature Box 2 -->
<!-- Feature Box 3 -->
<!-- Feature Box 4 -->
</div>
<!-- Row Ends -->
</section>