Yeah, there has been a bit of a problem with gmail in particular and the newest update. Everyone had to authenticate in safari which opens a dialogue box and then allows you to authenticate. Unfortunately, that hasn’t worked for a lot of people. A couple of my gmail accounts randomly go offline, but I can always get them back on. Several people aren’t so lucky.
Unfortunately, we are just waiting for the bug fix here so your options are use another method for checking email until it is fixed or roll back to the previous version of Mojave.