Designating your blog as private, to be read by authorised parties only, was a much requested feature.
Blogger uses your Google account to identify you, and verify that you are authorised to view any blog (including yours). And that causes occasional concern, fortunately unnecessary in this case.
Seeing your Google account, which coincidentally may be the same as your email address, as you peruse your blog, is a bit scary.
As you see your email address in your Blogger page header, every visitor to your blog is seeing their own email address in their Blogger page header.
Everybody sees their own email address - nobody sees yours, except you.
Nobody else sees your email address. I see mine, where you see yours.
A recently seen increase in spam is not from your email address being visible.
If you're seeing an increase in spam recently, it's not related to your Blogger / Google account being visible to spammers.
- Improved spam filtering requires increased spam volume, in order to allow the spammers to maintain a constant income level.
- Increased availability of bots makes the increased spam volume possible.
Work on important issues - this is not one to worry about.
Relax just a bit. Concentrate on other, more important issues. But accept the inevitable - you will need a Google account, to use Blogger. Your (Blogger) Google account, and your (GMail) Google account don't have to be the same, though. Google accounts are free, and you can set up another, if you like.
(Note 2/13): As noted by Pete in his separate comment, your email address is kept private, by carefully designed code:
The iframe separation keeps arbitrary Blog*Spot blogs from reading (for example) your Google Account e-mail address when you're logged in.