There are three possible causes for the sunvisors faling down. Yes, the screw can loosen. If that is the case, tightening the screw will solve the problem. The screw tightens a plastic sleeve onto the pin that it rotates on. If that sleeve is broken or missing, then tightening the screw will have no effect. Replace the sleeve. I've attached a link:
Sun visor sleeve.
The third possibility is that the pin itself is loose in the cast base piece that screws to the a-pillar. To check for this undo the screw and slide off the sunvisor. See if the pin you've exposed is fixed in place (good) or rotates n the base (bad). If it rotates you have two options. You can epoxy it in place or replace the entire base. The base for a t-top or factory convertible is not the same as used on a hardtop car.
Lon