Can't reply to comments in Blogger?

You'll need to enable third party cookies.

Update:

In Google Chrome, you can add an exception for third party cookies in Privacy > Content settings, rather than allowing all.

In Firefox I found that it was Privacy Badger causing the problem by blocking the blogger.com cookies. Moving the slider to Allow fixed the problem.

I had already allowed third party cookies in Firefox, because blocking them caused a problem on a particular website (can't remember where), but it is also possible to block them and add exceptions in Preferences >Privacy > History > Use custom settings for history.

Update 2:

If you have blocked third party cookies in Firefox, add an exception for https://blogger.com:

Note: https