If I play a damage spell to an enemy unit at a battlefield, can the opponent use a reaction spell that moves the unit to base, hand scot-free?

Yes, your opponent can use a [Reaction] spell to move the targeted unit to their base, which will cause your damage spell to fail to deal damage to that unit.

According to RiftJudge FAQ #3011 and FAQ #5858, here is how the interaction works:

The Sequence

1. You play your damage spell: You choose a target (an enemy unit at a battlefield) and pay all costs. The spell goes on the chain.
2. State becomes CLOSED: Because there is an item on the chain, players can play [Reaction] cards.
3. Opponent reacts: The opponent plays a spell that moves their unit from the battlefield to their base. This reaction goes on top of the chain (above your spell).
4. Chain resolves (LIFO - Last In, First Out):
   • The opponent’s spell resolves first, moving the unit to their base.
   • Your damage spell then resolves.

The Result

When your damage spell attempts to resolve, it checks if its target is still legal. Because the unit is no longer at a battlefield, it is an illegal target. As per RiftJudge FAQ #3011:

• The spell does not "fizzle" in the traditional sense; it still technically resolves, but the targeting effect (dealing damage) cannot happen because the target is invalid.
• Any other effects on the spell (such as "Draw 1") will still occur, provided they do not require the damage to have been dealt.
• You cannot choose a new target. Once a spell is played and targets are locked in, you cannot change them if the original target moves or becomes invalid.

In short, while you do not get to "counter" your own spell, the opponent successfully saves their unit from the damage by moving it to an invalid location.