There is a showdown on the battlefield Altar of Blood. At the end of this showdown, units from each side are about to die from damages. Which player chooses first to pay the 3 power cost to save their unit?

Ruling: The turn player chooses first to pay for their units, then the other player in turn order.

Here's why: At the end of combat, all units with lethal damage die simultaneously during the Cleanup (Rule 323.5). Altar of Blood creates a replacement effect for each dying unit — "its controller may pay [A][A][A] to heal it, exhaust it, and recall it instead." Each unit's death is a separate event, and each player decides for their own units.

Per Rule 373.1, when replacement effects from different controllers apply to simultaneous events, they execute in turn order. So the turn player makes their pay/decline decisions first for their dying units, followed by the other player.