Although a fire source (or the ether medallion) can be used to reveal the invisible bridges in this room, this is typically never done as there's a simple straight-line path from the portal to the door.

The fastest way through the room is simply to dash left, killing the blue bari along the way. Assuming you entered the portal from its top edge, you can technically save about 2 frames by moving downward to align with the door before dashing. However, this is not recommended as it increases the risk that you'll be too low to hit the bari but high enough to still be hit by it, costing you over 4 seconds.

Note that without a sword, you'll actually want to take the opposite approach and move further downward, to avoid contact with the bari entirely.

Without the boots, you'll simply have to dodge the bari while walking left and attempting to align yourself with the door.