Does damage prevention apply before or after deathtouch is checked?

Damage prevention happens during the damage-dealing process itself, before the game checks the results of that damage. According to CR 120.4, if damage would be dealt and a prevention effect applies, that damage is prevented first — the prevented damage is simply not dealt. Only damage that is actually dealt can have results like loss of life, destruction via deathtouch, or marking damage on creatures.

Deathtouch is a static ability checked after damage is dealt (CR 702.2c, 120.4). A creature with deathtouch destroys any creature it deals damage to, but the key word is "deals." If the damage is fully prevented, no damage is dealt, so deathtouch has nothing to act on. The creature that would have been dealt deathtouch damage survives unharmed.

If the damage is only partially prevented, deathtouch still applies to the remaining damage that was actually dealt. Even 1 point of deathtouch damage getting through is enough to destroy the creature, because deathtouch states that any amount of damage dealt by that source is lethal (CR 702.2b).

Concrete example: Your 1/1 blocks an opponent's 2/2 with deathtouch. You cast Fog to prevent all combat damage this turn. Because all damage is prevented before it is dealt, the 2/2's deathtouch ability never applies, and your 1/1 survives.

Similarly, if your 1/1 had a "prevent 1 damage" shield and the deathtouch creature dealt exactly 1, that 1 damage is fully prevented — no damage is actually dealt, so the 1/1 is not destroyed. Partial prevention still lets deathtouch function on any unpreventable remainder.