The nature of covalent bonds is such that all of the valence electrons are either bound to an atom or shared in bonds. Consequently there are no free electrons that can easily move between atoms as there are in metals. Therefore, generally covalent crystals will not conduct electricity in their solid form.
Many covalent crystals will not dissolve in water, consequently those classes of materials will not conduct electricity when placed in water eg.: diamond). However, there are some covalent crystals that will dissolve in water (eg.: sugar). Again because the electrons are bound to specific molecules and not allowed to move through the aqueous solution as the ions of soluble ionic crystal can, these molecules will not be electrolytic. That is, they will not conduct electricity.
Melting covalent crystals do not separate the individually bound atoms in the covalent bond, consequently there are still no lose electrons to conduct electricity.