I don't think anyone died on Enterprise for the first two seasons. In fact the first dead on board were in the episode Damage, if am not mistaken, toward the end of season three.
No, the first casualty was Crewman Fuller in "Anomaly," the second episode of season 3.
Crewman Novakovich was originally supposed to die in the transporter accident he endured in the third episode of the series, "Strange New World." But the producers didn't want to repeat the previous shows' pattern of trivializing death by having someone die in act 2 and then end the episode with the characters laughing and cheerful, so they added a line saying that Novakovich had survived his injuries. After that, they didn't want to kill off a member of the crew until they were able to give it the emotional weight and impact it deserved.
For all the (often deserved) criticism ENT gets, this is one thing it did far better than any other Trek series: approaching death with suitable maturity and seriousness. They didn't kill off crewmembers as a cheap plot device, and the deaths that did happen in seasons 3 and 4 had real meaning and consequences to the characters.