Speaking to YouTube channel Back2Warcraft, Blizzard senior producer Pete Stilwell said the team was "hoping to have [the beta] start at some point early next year". The developer has not set an exact date but "the sooner we can get it started the better", he said.
The beta will be more like a traditional "pre-alpha" than a normal beta, he said, with features tested as soon as Blizzard finishes them. For example, when the Undead race is complete, the beta might host 1v1 matches "so that there can be a really good feedback loop, so that if we need to make any course corrections, that can happen". He said he did not know how long the beta would last.
Blizzard has already launched a web page with instructions on how to make yourself eligible "for a chance at being selected" to play the beta.
You can watch the interview below—Stilwell's comments start at 14:57.
Warcraft 3: Reforged is due out at some point before the end of 2019, so having a beta at the start of the year would give Blizzard plenty of time to make tweaks. The remaster includes The Frozen Throne expansion, and if you buy it you'll be able to play multiplayer games against people with the original, non-remastered version.