Chaver: Yes if you’re invited to his/her half birthday party. Yes if you’re his/her mom or dad.
All other people, including nannies and babysitters – it depends.
As for whether the present is full or half…
Most likely, the present will be full, even on a half birthday. But there are significant exceptions, especially when it comes to monetary gifts.
If a full (regular) birthday gift warrants a sum of $50, then perhaps a half birthday gift could be $25. The same rationale goes with gift cards.***
***BUT: If the birthday sum given is a chai denomination, such as $18, $36, or $54, then please consult with your orthodox rabbi to determine the proper halving halacha.
Thank you 🙂