Density of the proton wind coming from the Sun is few ten particles per cubic centimeter (atmospheric density is orders of magnitude higher in LEO), thus not influencing a satellite by drag. Actually, the proton flow could also add tiny amounts of speed to the satellite when hitting it from the back.
The solar storm is accompanied by a tiny magnetic field (up to 10 nT, less than Earth magnetic field) which could cause some electrodynamic drag, but not very much.
There are two possible ways how increased solar activity can degrade a orbit in indirect ways:
First, the protons (as far as they do penetrate the Earth magnetic field at all) can cause radiation influenced damage to on-board electronics. This could lead to a failure of the guidance system and subsequently to unplanned orbital maneuvers. Fortunately, these systems are usually redundant and therefore save from such errors.
As you explicitly mention Skylab - its faster reentry was due to increased solar activity over a longer period. The activity heats up the upper atmosphere and increases its density. Some nice plots can be found in this paper: Empirical Modeling of the Thermosphere.
As you can see, solar activity can change the density of the medium in LEO by more than a factor 10. Any increase in density proportionally increases the drag and decreases orbital lifetime. It seems that in the Skylab case the estimate of atmospheric density was about half of the actual density, therefore halving the time before a reboost of the station was necessary.