The magento downloads portal does not have a SUPEE 6788 patch for 1.11.
Any advice?
Magento CE 1.6 / EE 1.11 is delayed for about a week per this => https://community.magento.com/t5/Security-Patches/Magento-1-6-SUPEE-6788/m-p/20064#U20064
Patches for 1.5.x.x and 1.6.x.x are online, check with your EE customer support as 1.10/1.11 should be online for you as well.