It's not clear.
We know that each Magic Missile dart strikes simultaneously (from the spell description), and we know that damage is rolled only once for all of them. But, we also know that each dart is a separate source of damage. So, our question really boils down to this: Does a death saving throw failure happen for each time that a 0 hp creature is dealt damage, or for each source that deals damage?
While there's isn't a direct official ruling on how multiple Magic Missile darts affects death saving throw failures that I could find, there is a ruling on a similar question: If you are concentrating on a spell, and get hit by three Magic Missile darts, do you need to roll a concentration saving throw for each of them? Per a Jeremy Crawford tweet,
Concentration: "You make a separate saving throw for each source of damage" (PH, 203). Roll for each missile.
The rule referenced there (I'll quote from the online Basic Rules):
Taking damage. Whenever you take damage while you are concentrating on a spell, you must make a Constitution saving throw to maintain your concentration. The DC equals 10 or half the damage you take, whichever number is higher. If you take damage from multiple sources, such as an arrow and a dragon’s breath, you make a separate saving throw for each source of damage.
So concentration saving throws specifically say to look for each source of damage, which is the basis for Jeremy's tweeted ruling. Now that I've gone on that tangent with concentration, let's compare that wording to the wording on taking damage while at 0 hp:
If you take any damage while you have 0 hit points, you suffer a death saving throw failure. If the damage is from a critical hit, you suffer two failures instead.
There are two ways one can look at this:
- "If you take any damage" (from the death saving throw rules) and "Whenever you take damage" (from the spell concentration rules) are equivalent wordings of the same concept. (Some games like Magic: the Gathering distinguish between meanings of "If" and "Whenever", but D&D is written in a more "plain English" style and nuances of which word starts a sentence is not as important.) In that case, we should use the precedent from Jeremy's tweet as an indication that yes, as each dart is its own source of damage, and rules that refer to taking damage look at each source separately, therefore each source causes its own death saving throw failure.
- The "Whenever you take damage" paragraph from the spell concentration rules specifically calls out in its own sentence that it applies to each source of damage separately, while the analogous "If you take any damage" paragraph from the death saving throw rules specifically does not include any such description of how it applies to multiple sources. Therefore, no, it only applies once to each time you take damage, as the "trigger" is just "if you take any damage", and you just take damage the once even though it's from several darts. If it was supposed to apply to each source separately, it would have said so explicitly like the concentration rules do. (This reading of the situation is similar to the real-world law interpretation principle of "Congress is presumed to act intentionally and purposely when it includes language in one section but omits it in another.")
But unless we get further official guidance, we don't really know which way it is "supposed" to be, and I've already analyzed the text much more than was probably intended.
So ask your DM
As with all the rules, reading and interpreting them is up to your DM. If this scenario is likely to come up in your games, it's best to work out how your DM will rule this case ahead of time.
And if you're the DM, just go with whichever interpretation you think is most correct or will lead to the most fun gameplay, and ensure your players know how your game will work.