Can you respond to Mindslaver's activation before your opponent gains control of your turn?

When a player activates Mindslaver's ability (by tapping and sacrificing it), that activated ability goes on the stack like any other ability. Both players receive priority in succession before it resolves, meaning you absolutely have the opportunity to cast instants, activate abilities, or otherwise respond before your opponent gains control of your turn (CR 116.2b, 116.3).

The key point is that Mindslaver's effect — controlling your next turn — only happens when the ability resolves. Until that moment, the game is proceeding normally under standard priority rules. You are free to use any instant-speed spells or abilities in response (CR 117.1).

Note that once the ability resolves and your opponent's control of your turn begins, you lose the ability to make decisions that turn — your opponent makes them for you. So responding before resolution is your last clean window to act freely (CR 714.1).

Example: Your opponent activates Mindslaver. In response, you cast Counterspell targeting the Mindslaver ability. If Counterspell resolves, the Mindslaver ability is countered and your opponent never gains control of your turn.

Even if you don't counter it, you could use that response window to activate a planeswalker ability, cycle a card, or crack a fetch land — anything at instant speed — before surrendering control of your turn.