No, governments should not monitor civilian email for terrorist activity. This is a gross violation of citizens' right to privacy. A citizen has the reasonable expectation that a private communication stays private. For the government to spy on these private communications flies in the face of that reasonable expecation and should not be permitted.
A very small part of me feels that maybe they should run bots and if anything is suspect it could be investigated further, but it is a slippery slope. For the most part, I feel that people should be entitled to their privacy and feeling that no one will see their conversations but themselves and the recipient.
The 4th Amendment is very clear on this point, we should be secure in our papers and effects. That applies to emails. We know from reading some of the founders' writings that they intended it to apply in the future even as technology changes. So almost everything the NSA does is unconstitutional.