By default “Remember me” functionality is not enabled in magento.
To enable it, login to admin panel then
Go to System->Configuration->Customers->Persistent Shopping Cart->General Options->Enable Persistence
From here select “Yes” options from select box and click on “Save Config” button.
Now refresh login page and you will see “Remember Me” along with checkbox will appear on login page.
If not appear it on login page then clear your cache from Cache Management.