I am in no way advising you to do anything I would do but... if it were me.... I wouldn't password protect the computer, though it's easily done by putting a drivelock password on it, so it doesn't even boot otherwise. HOWEVER.. I wouldn't! Doing so, she will more than likely feel attacked. That would be bad. She will probably pull further away and in the end, it will not benefit you or your kids at all.
I have parts of the same issues, husband pulling away, refuses communication, refuses medication... But haven't yet figured out exactly how to deal with his specific situation, as I have just recently started trying and becoming determined to get him help. But, typically people who have those problems go to online activity because it isn't the "real world", it's a place where she can be who she wants to be, no one knows her life, no one knows what kind of person she is and she can choose to make herself out to be a total victim if she wishes to do so. She holds control over how people see her. In her real life, she's depressed, she feels like she has no control and wants to block herself out from the rest of the functioning world.
All you can do is try to support her recovery. Reassure the kids that mommy and daddy both love them and that everything will be just fine. Just remember when caring for a spouse with issues like this, you have to factor in the kids and how it will affect them! Try your best to go out of your way for them NOT to see her telling you she doesn't love you or that she wants to leave or escape. They may not understand that it's just because something is wrong and may read into it in ways they shouldn't.
Again, I started to ramble a little and got off of what I was trying to say.
Support her, try to encourage her therapy and talk to her doctor about her not taking her medication. Try to explain to her that even if in the end she does leave you for whatever reason, you just want to know that she is well, for her sake, yours and that of your children and only want her to take the medication advice of the doctors for that. Tell her that you too are considering therapy or a medication for depression, even if you aren't. Try to make her feel like she isn't in it alone. Tell her that you would like for the two of you to get help together for the sake of the family and raising your children in a home that isn't broken. I'm sure doing everything is taking a toll on you anyway, so it all falls into place.

That may or may not be good advice but... again.. it's what I'd do....
Bottom line. Act. But Don't act harshly. You've done well by having that level of tolerance. Most people wouldn't. Kudos to you for holding out the way you have and stepping up to be a good dad/husband when it's truly needed.