Considering that a Monk is Proficient with Shortswords and Daggers, both Finesse weapons, they could by raw use Sneak Attack.
Martial Arts allows you to use Dex for both attacks and damage on unarmed strikes, and all monk weapons. That's basically Finesse. So I would allow it, especially when considering the highest damage die for a monk weapon is a d8, which matches up with a Rogue's Rapier. Most are d6s, matching the shortswords. If you can get sneak attack using a Dagger or Shortsword, there isn't much wrong with using fists, which deal basically the same damage.
Like others have said the only thing you have to consider is that flurry of blows/extra attack allow you more chance to land a sneak attack. But one costs ki, so you won't use it all the time and your sneak attack dice aren't going to be as high as a rogue's since you are multiclassed, so I think it balances out.