The core Recent Posts Widget does not output any classes for the current post - probably because the Widget semantically is a list of posts, rather than a navigation menu, and therefore has no intended purpose to reflect current location.
That said, with a bit of CSS/jQuery trickery, you could target the current post by matching the current post's permalink and/or ID to the href and/or title attribute of the Recent Posts Widget list item link.
While the implementation itself is probably outside the scope of WPSE, the relevant code is here:
<?php while ($r->have_posts()) : $r->the_post(); ?>
<li><a href="<?php the_permalink() ?>" title="<?php echo esc_attr(get_the_title() ? get_the_title() : get_the_ID()); ?>"><?php if ( get_the_title() ) the_title(); else the_ID(); ?></a></li>
<?php endwhile; ?>
Use jQuery to match e.g.
.widget li a[href=<?php get_permalink(); ?>], or something along those lines.