By default “Remember me” functionality is disabled in magento. You can enable it form Admin Panel

Follow these simple steps to Enable this functionality:

System->Configuration->Customers->Persistent Shopping Cart->General Options->Enable Persistence. From here select Yes from select box and click on “Save Config” button. Now refresh your site and you will see “Remember Me” along with checkbox will appear on login page. If not then clear your cache from Cache Management.

Third Party Extension

Extension Name: Remember Me

Extension Link: http://www.magentocommerce.com/magento-connect/remember-me.html