Use this custom CSS rule:

div.ecwid-productBrowser-auth { 
  display: none;
}

This code will remove the "sign in" link.

If you also want to remove all elements, that allows signing into your store, use this code: 

div.ecwid-productBrowser-auth { 
  display: none;
}

table.ecwid-Checkout-blockTitle button, div.ecwid-Checkout-PasswordBlock {

  display:none;

}

div.ecwid-Checkout-blockTitle button.gwt-Button{

  display: none;

}

Then edit the labels  that states "Enter your shipping address or .." and "Enter your e-mail or .." on the checkout pages.