In the context of the film itself, the Covenant crew felt safe because:
1. They followed a complete stranger after being attacked by a hostile alien lifeform.
Largely because they had no choice. Their ride was a burning pile of rubble. Remember?
2. Into a giant alien city which had no doors.
It did have a door. One that could be closed behind them and made secure. A gigantic friggin' round one.
3. Said giant alien city was found to be filled with corpses.
Once again, they had no choice.
4. After being asked if the place is safe, the suspicious stranger told them that it was.
Which it was, as far everyone was concerned. At the very least safer than being outside. We don't know if David was aware of the Neomorph's ability to scale walls. For all we know, he was telling the truth. His attempt to tame the creature may have been the first time he's encountered one inside the building alive that wasn't one of his own experiments.
Would you feel safe if a stranger took you hiding in a doorless alien graveyard?
Feeling safe has nothing to do with it. There was no other choice. 'safer', sure. Truly secure and ready to sit down watch some sitcom with their new found buddy..? Maybe not.
-Windebieste.