I think there are multiple aspects to this question.
If I'm not mistaken, your concept would basically mean, that every floor is anchored to those 4 cables in the corners, thus basically "floating" in the air?
As a consequence, the upward force generated by those balloons would be (greater) equal to the weight of the floors. Something like this:
[![enter image description here][1]][1]

In this graphic I have neglected the tension due to the dead load of the cables.

**stability and safety**

Probably the biggest problem aside from statics would be the stability and safety of the structure; leaks, failure of a balloon, wind, etc.

**statics**

In normal conditions, Helium has a density of $\rho_{He}\approx0.179kg/m^3$, while air's is around $\rho_{air}\approx1.29kg/m^3$.
If we neglect the weight of the hull of the balloon as well, we can come up with a net buoyancy force $f_B$ per cubic meter of Helium.
$$ f_{B}=(\rho_{air}-\rho_{He})g=10.9N/m^3_{He} $$

If we take a $10m\times10m$ concrete floor, with a thickness of $t=0.2m$, and a density $\rho_c=2'500kg/m^3$, you would already need a balloon volume
$$ V_{He}\approx45'000m^3 $$
which is a sphere of radius $r=22m$. An that is for one single floor.

**availability of helium**

Though being the most abundant element in the universe, helium is relatively rare on the earth. Currently, most helium is extracted from natural-gas, but those fossil fuel resources are limited. I'm not sure whether we earnings would want to risk losing such a precious resource.

  [1]: https://i.sstatic.net/iZbc9.png