One of the most effective ways to deal with a mini-sentry is the Red-Tape Recorder.
When you place a sapper on a regular sentry, most engineers will try killing you first, then proceed to repair buildings. Mini-sentries, on the other hand, are considered disposable and usually are much more easier to destroy if damaged or out of ammo, and rebuild.
The Red Tape Recorder addresses both these issues. WHen a sapper is attached to a sentry, the engineer cannot use his Destroy PDA to destroy it, and he must wait for the sapper to either A) be knocked off or B) kill his sentry. The beauty of the red-tape is that it has a ridiculously long "destroy" time. I put destroy in quotes because the RTR instead reverses construction of buildings.
The time it takes to kill a minisentry with a sapper is 3 seconds. With an RTR though, it takes 10 seconds. This means this is 10 seconds an engineer is without a sentry, and might be forced to retreat.
Other than that, for other classes, using melee weapons like the Homewrecker for Pyro tends to work extremely well. For soldier, using the direct hit is an almost guaranteed insta destroyal of a minisentry. Scouts can easily take them out from long range using a pistol.