Can I react to someone playing a unit to their base from their hand?

No, you cannot react to an opponent playing a unit to their base.

According to RiftJudge FAQ #7317 and Rule 359.3, when a player plays a unit, it is placed on the chain as a pending item, but it is removed from the chain immediately upon finalization. Because units resolve immediately and do not linger on the chain, players never gain priority to respond to the act of playing the unit itself.

Key points:

• No Reaction Window: There is no opportunity to play a [Reaction] spell or activate an ability in response to the unit being played.
• Triggered Abilities: While you cannot react to the unit being played, if that unit has a "When you play me" triggered ability, that ability will go on the chain after the unit enters the battlefield. You can react to that triggered ability once it is on the chain.