We get this question a lot so here is a good walkthrough on how to get this dressphere:

CH2 - In 'Infiltrate Bevelle' mission - when you enter screen labeled 'Bevelle - labyrinth' for second time (this is after the screen with gaol machine), pay attention. Approach the wall which will fall away. Ahead you see three lifts. Use left one first - on upper level climb both elevations to get access to final lift up here (open the chest for HI-POTION). Now go S all the way until you fall down. Use N lift now to be transported to elevated switch - climb onto it and two pillars will rise somewhere. Go back down, and now ride right lift to upper level. Take left lift back down to lower level. Use N lift again and you get access to another elevated switch - climb onto it and you have now complete access. Use the lift to ride back down, then use left lift to go back up to upper level. Finally use top lift here to get even higher. Up here use the pillars you've been rising to nab this DSP.

CH3 - If you've sided with New Yevon, you have permanent access to Bevelle's dungeon thru whole CH. Just follow steps given above and get this DSP.

CH3 - If you've sided with Youth League, the access to Bevelle's dungeon is available only AFTER you clear Besaid and Kilika hotspots, but BEFORE you waste the boss inside Djose temple. Just follow steps given above and get this DSP.

CH5 - Regardless of who you sided with, you have permanent access to Bevelle's dungeon. Just follow steps given above and get this DSP.

Credit: gamefaqs.com