There is no specific date listed, so let's take a look at all of the clues to try to get a range of time that fits.
TL;DR: the events of 'Betrayal at Baldur's Gate' likely take place between the years...
- 1022 DR ----> 1222 DR or 1235 DR ----> 1358 DR or 1482 DR ----> Present
... but may also take place between the years...
- c. 261 DR ----> c. 661 DR or 720 DR ----> 1021 DR
Baldur's Gate
The game is set in Baldur's Gate, so at the very least we can narrow the timeline down to the years during which Baldur's Gate existed.
We don't have an exact founding date for Baldur's Gate, but the first dated event is referenced in Baldur's Gate: Dark Alliance for the PlayStation 2. When the heroes in that game meet the reincarnated Eldrith, they can learn of an event in 1235 DR. This will be a potential start date for something to work with, but is hardly definitive.
Baldur's Gate has not fallen, so we lack an end date for our range.
Bhaal
The source of conflict in the board game is described in the rulebook:
The shadow of Bhaal has covered the city, and now the god of murder is whispering to those who would carve a path of bloody mayhem in his name.
Because we know that Bhaal is present, we get a few more ways to narrow the timeline.
We don't get an ascension date for Bhaal originally (although Faiths and Avatars describes the ascension):
During [ages past], there arose three powerful mortals — Bane, Bhaal, and Myrkul — who lusted after the power Jergal wielded... Eventually the trio destroyed one of the Seven Lost Gods, and they each seized a portion of his divine essence for themselves.
We still lack a definitive start date, but Bhaal does offer us some insight into end dates. During the Time of Troubles in 1358 DR, according to Lost Empires of Faerun...
As fate would have it, however, the god of murder was himself murdered by Cyric during the Time of Troubles.
... he was slain not resurrecting until more than a century later in 1482 DR, during the Second Sundering. The events leading to this are detailed in the adventure Murder at Baldur's Gate (spoilers below):
The characters can’t know it at this point, but the death of the slayer has completed Bhaal’s resurrection.
This places our current timeline restrictions as
- Ages Past (tentatively 1235 DR) ----> 1358 DR or 1482 DR ----> Present
The Harpers
The rulebook also states:
You have been called upon by the Harpers to stop the evil that has infected the city.
There have been many incarnations of the Harpers, only some of which exist within our current timeline restrictions. The original Harpers were a group called the Harpers of Twilight conceived by the leaders of Myth Drannor around the year 261 DR, according to The Code of the Harpers:
[The wise ruler of the elven realm] decided that welcoming them was the best policy for the elven race.
[...]
By the time the Mythal was laid and the city of Myth Drannor opened for dwarves, gnomes, halflings, humans, elves, and their various halfbreeds to dwell together (DR 261, the Year of Soaring Stars)...
[...]
An idealistic and powerful elven mage, who took the name "Lady Steel", agreed to head this [secret organization that was needed to work for the causes of good]
[...]
they became known as the Harpers at Twilight.
From there we get a number of dwindlings and reemergences for the Harpers. The Code of the Harpers describes most of them (others are described in The Grand History of the Realms), and I will list them all below (removing the period of time where Bhaal was awaiting ressurection):
- c. 261 DR to c. 661 DR
- 720 DR to 1021 DR
- 1022 DR to 1222 DR
- 1235 DR to Bhaal's murder
- Bhaal's ressurection to Present
The restricted timeline with this information is as follows:
- c. 261 DR ----> c. 661 DR or 720 DR ----> 1021 DR or 1022 DR ----> 1222 DR or 1235 DR ----> 1358 DR or 1482 DR ----> Present
Elfsong Tavern
The next tidbit we get is the existence of Elfsong Tavern (from the Rulebook):
The city is different each time you step out of the relative safety of the Elfsong Tavern.
Elfsong Tavern is hard to pin down on the timeline, but we get some insights in Volo's Guide to the Sword Coast:
This tavern is named for an unusual haunting — a ghostly female elven voice, heard from time to time all over the establishment.
But we don't get much more information to tether the tavern or this woman to any part of the timeline.
(I saw some tidbits about the Onyx Tower and some Company of the Crescent Blade, but couldn't find a source to reinforce them, and found a few contradictory sources. I also found a quote from Eric Boyd regarding the College of Cli, but it was explicitly mentioned as only his notes and not necessarily canon)
That's all we've got
With so little to go off of, there isn't much else we can determine. With some educated guesses, I'd wager that the earliest incarnation of the Harpers that makes sense is 1022 - 1222 version, but that's just a guess considering the prior versions didn't have the level of influence that the ones beginning in the 11th century DR had.
My final judgement would be that the events either occur between...
- 1022 DR ----> 1222 DR or 1235 DR ----> 1358 DR or 1482 DR ----> Present
