According to the Codex entry for kinetic barriers:
Kinetic barriers are repulsive mass effect fields projected from tiny emitters. These shields safely deflect small objects traveling at rapid velocities. This affords protection from bullets and other dangerous projectiles, but still allows the user to sit down without knocking away their chair.
The first thing this does is explain that the barrier has a deflection effect on matter. Although I'm not a master of theoretical physics, this could be accomplished by using a mass effect field to give the wearer "negative" mass at a precisely calculated point during the projectile's flight. Such a negative mass would cause a negative gravitational effect (imagine if the sun's gravity (a product of its mass) repelled the Earth instead of attracting it).
It would seem that the barriers are somehow active, not passive, since a simple mass effect field cannot differentiate between the mass of a chair and the mass of a bullet. The presence of multiple "tiny emitters", which would lead me to believe that the emitters are focusing and finely controlling the mass effect field's effects to specifically target and deflect small incoming projectiles (especially since the barrier doesn't "repel" the ground with its mass effect fields and cause the wearer to go flying up into the air every time they are shot).
Computerized Targeting and the "Reverse Firing Solution"
From the Codex entry for the M-622 Avalanche:
[It can cause] great damage to electronics like kinetic barrier emitters, which sometimes leads to total systems failure.
It would seem that the emitters are some sort of electronic (probably computerized) system, and it seems to work together as a unit (since "total systems failure" is an option). It's reasonable to assume that in a world where FTL jumps can be quickly and accurately plotted by computers, a computerized system should be able to detect incoming mass of whatever size is specified moving at whatever velocity is specified, and then calculate a "reverse firing solution" to deflect it before it can cause damage (yes, I made up that term just now).
Another thing that the codex entry for this weapon points out is that energy (such as that from a weaponized super-cooling effect) is likely unhindered by barriers, which makes sense logically, since energy has no mass for these systems to affect. But that is neither here nor there.
But what about…
The weapons in the series use mass effect fields to propel small chunks of metal to extremely high velocities, so how could even an awesome futuristic computer detect and calculate it that quickly? That I couldn't tell you. They have quantum entanglement communicators, so maybe they have extremely powerful quantum computing technology small enough to be outfitted to personal armor.
So assuming all that, my guess as to why melee attacks cannot be blocked by barrier shields is that the system is tuned that way on purpose. I.e.: Both the mass and velocity of the object must be within specified parameters in order for the system to activate the deflection effect. This would prevent the wearer's arm from getting ripped off if they stuck it through the barrier's "event horizon" of deflection (which is another indication that the system is active, not passive), as well as allow them to run/sprint without nearby objects being deflected randomly by the barrier.