This is a slippery slope that I’ve slid down before.
A now ex-girlfriend of mine told me her e-mail password (without me expressing any desire what so ever to know it) and then got pretty offended when I didn’t share mine. I think privacy is an important part of a successful relationship and told her so. This brought up the question “Well, what do you have to hide?” I didn’t really have anything to hide, but the idea that she felt the need to spy and peek in on my correspondences to validate the trust she supposedly had in me felt wrong.
I, personally, would never ask my significant other for their e-mail password, and in a situation like the above, if I knew it, I wouldn’t use it. Nor would I give them mine.