The important thing is that you're adding the Suffocate Markers to the Drown enchantment rather than to the creature. Thus, it is the enchantment that is being affected, and the controller of the enchantment that gets to decide when the markers are added. So they could indeed choose to add the Suffocate marker before paying the upkeep cost. Whenever the Suffocate marker is added, the creature loses 2 Life.
This is probably most similar to Death Link where a single effect seems to be affecting objects controlled by both players. Just like with Death Link, it's important to recognize that these are not two separate effects though. Adding the Suffocate marker and the creature losing 2 Life are a single effect that can not be separated from each other. They have to happen simultaneously. You can't add the marker, pay the upkeep, and THEN have the creature lose life. So, if the controller of the enchantment adds the Suffocate before paying upkeep, there is no recourse for the creature's controller to prevent destruction (barring other effects like healing or life gain).