Density functional theory optimizes the total energy with respect to the (possibly constrained) electronic density. The usual numerical procedure for this optimization is embodied by the KS method. The KS procedure will give you the optimal total energy and the ground-state (gs) density for which this optimal energy is achieved. This optimization procedure is independent of the explicit exchange-correlation (xc) density (or possibly orbital, where OEP-KS method applies) functional provided by you. Although, of course, the better the xc functional, the closer the total energy and gs density will be to the true one.(see Note added to the end)
The minimization procedure is done in the (in principle) infinite space of real valued functions that represent the density. However, by choosing a finite basis you constrain this search space. The more complete your basis set, the more flexibility you have to find the gs density, because the density is built from the single particle orbitals expanded in the given basis.
With this reasoning, it seems plausible to think that if you increase your basis set, you will have a bigger search space and the optimal gs density will be better than for a smaller basis set. This behavior would be in general the expected. However, it seems to me possible that you could add more basis functions that could increase the flexibility of the density in high energy regions, while deteriorating the description of the density in the low energy region (here I mean regions of the infinite dimensional space of densities), so one could somehow arrive at a higher energy by increasing the basis set in the wrong way. For example, think of a 1D quantum well in a real space representation of delta functions centered in a dicrete grid of points. If you increase your basis by adding many delta functions in low density regions but remove a few in places where the minimum of the potential occur, then for sure your total energy will increase, because your basis is incapable of describing accumulation of charge in the minimum of the potential.
Therefore, the take-home message is: if your energy is increasing upon increasing your basis set, then you are adding flexibility in the wrong place and deteriorating in the right place. If this is the behavior, then try to increase your basis in a different way, until you see a lowering of the total energy.
Note: All above is valid for a given fixed xc (explicit) density functional. Changing the functional will change your energy in a non-variational way. I mean, a given xc functional could give a total energy lower than the true one. However, this does not mean that there is a problem in the KS minimization procedure, but that your xc functional is not a very good one.