I just bought and installed Diablo II: LoD yesterday (reinstalled again today to try and fix the issue) and when I try and log in it just gets stuck at the “Checking versions…”
You can run through this support article to see if any of them help with the issue.
I have a few friends that I’ve helped get Diablo 2 running on… After installing the game using Blizzards downloader, they were also stuck at the checking versions screen. Manually downloading and installing the latest patch was what fixed them up, so that may be the ticket that solves the issue for you.
One thing worth noting is the usage of a VPN, an internet proxy and/or while connected to a business-class internet service is not allowed while playing Diablo 2 on Battle.net. Attempting connect to Battle.net with any one of them being true will result in an automatic two-week restriction. Subsequent attempts with any of them being true resets the duration.
@Blizz: It would be a good idea to update the downloader to download a fully up to date version of the game. 
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Having the same issue here, please someone at @Blizzard can you check for Account or IP bans on your side for both DireRaven and I? I’ve already been through your list of connectivity troubleshooting recommendations. I suspect the VPN issue is the cause, but I was told to try again after 2 days previously but TheDarkJedi’s comment indicates that will reset some invisible timer for 2 weeks, regardless of my VPN being turned off now.
As long as you are not using a VPN, a proxy internet connection or are using a business class internet service when attempting to connect to Battle.net, then the timer will not be reset.