A quick fix would be to Duck Tape the base to the table!

A better idea would be to find a piece of steel either quarter inch or thicker and attach ( easiest way is to use epoxy )your present base to it and put some nonskid material on the bottom of the bottom the combined pieces .You can even customize the base by painting it and glueing refrigerator magnets upside down to the top to hold hooks! Another useful
Item you can add to you vise stem is a Cliffs shelf it not only holds hooks or completed flies it is also a hackle gauge.

Fishin' Jimmy