You are using Email to Case Premium and are curious as to if you can utilize the application within a Customer/Partner Portal or Community.
The E2CP new comment page performs 3 operations at once: 1) adds a comment 2) updates Case fields and 3) sends an email which gets stored in the standard Email Message object that Salesforce provides.
The Customer Portal/Community and Partner Portal/Community license types do not have Create access on the Email Message object and therefore an error will occur:
Sending e-mail with new comment failed: Insert failed. First exception on row 0; first error: INSUFFICIENT_ACCESS_ON_CROSS_REFERENCE_ENTITY, insufficient access rights on cross-reference id: 
Please vote up this Salesforce idea: https://success.salesforce.com/ideaView?id=08730000000aIIIAA2 as well as this Internet Creations' idea: https://ic.secure.force.com/myaccount/PortalIdeaDetail?id=08740000000TQHr
Additionally, other Email to Case Premium components such as the included fields and Case Presentation Layer will not be available to portal/community users as they would require a license to the application. While application licenses can be assigned to portal/community users, it would likely not be practical to assign licenses to your entire customer base.