How do I remove the handle of this utility sink faucet?
This type of knob has an internal splined hole which fits over the splined end of faucet stem. The knobs are typically made of die cast zinc or aluminium pot metal and then chrome plated. They are then fitted to a brass faucet stem. The screw at the top center of the knob fastens into a threaded hole in the end of the faucet stem to hold the knob in place.

It has been my experience that over time corrosion will occur with pot metal inside the knob spline. Commonly white in color this corrosion actually seems to expand in size from the original size of the spline hole inside the knob literally locking the knob in place. As you are finding this makes the knob very difficult to remove. 

When I have had to remove such knobs I have tried the same Liquid Wrench as you did but multiple applications over a couple of days. I also seem to recall trying heating up white vinegar and applying that. A syringe is useful for this application. Ultimately I found that repeated rocking the knob back and forth with good force helps break the hold of the corrosion and allowing more of the solvent to enter the spline joint.

In the very worst case you may end up having to break or cut away the skirts of the knob to get access to the faucet packing nut under the knob in order to get the stem out. If you can find replacement stems then that is optimal but there is also the possibility to cut the rest of the knob shaft off the stem using a Dremel tool with a cutoff grinder disk.