How to display partial values of JSON Encode values

Use json_decode:

global $wpdb;
$project_member_details = $wpdb->get_var( $wpdb->prepare( "SELECT project_members FROM wpxa_orocox_project_members WHERE project_id = 603" ) );
$project_member_details = json_decode( $project_member_details, true );
print_r( $project_member_details['member_image'] );

If you still want the output to be JSON encoded at the end of it, you could then do something like:

$member_images = $project_member_details['member_image'];
echo wp_json_encode( [ 'member_image' => $member_images ] );